ROBYN ARIANRHOD 拥有广义相对论博士学位，在莫纳什大学教授数学；她目前是莫纳什大学数学学院的一名附属教授。她为感兴趣的普通读者撰写有关数学和科学的文章——她的书已被翻译成多种语言，并入围国家文学奖，她的文章发表在各种媒体上，包括《澳大利亚最佳科学写作》。

ROBYN ARIANRHOD has a PhD in the theory of general relativity and taught mathematics at Monash University; she’s currently an Affiliate in Monash’s School of Mathematics. She writes about maths and science for the interested general reader—her books have been translated into several languages and shortlisted for national literary awards, and her articles have appeared in various outlets, including Best Australian Science Writing.

“阿丽亚罗德的向量，是一部科学论述的杰作，读起来就像是一场令人愉悦的认知悬念之旅，它介绍了向量、向量的概括以及与之相伴的数学物理符号工具，所有这些都通过与惊人联系和应用密切相关的历史小插图描述而相互契合。”

‘Arianrhod’s Vector, a masterpiece of science exposition, reads as a welcoming cognitive cliffhanger tour of vectors, their generalizations, and their accompanying symbolic tools of mathematical physics, all dovetailing through germane history vignette accounts of astonishing connections and applications.’

——约瑟夫·马祖尔， 《时钟幻象：我们测量时间的神话》作者

―Joseph Mazur, author of The Clock Mirage: Our Myth of Measured Time

“向量是具有大小和方向的量。这一思想使物理学家和数学家能够以新的维度想象和描述世界。作者追溯了向量在过去 5,000 年中的影响，以及向量（和张量）为何至今仍然具有现实意义。”

‘A vector is quantity that has magnitude and, crucially, direction. This idea has enabled physicists and mathematicians to imagine and describe the world in new dimensions. The author traces the influence of vectors over the past 5,000 years, and why vectors (and tensors) are still relevant today.’

―书商

―The Bookseller

“关于现代物理学的演变有很多书：从牛顿到麦克斯韦到爱因斯坦再到量子理论。但很少有作者关注使这些物理理论成为可能的数学革命。只有当数学工具包从简单的标量扩展到包括四元数、矢量和张量等工具和思想时，物理学家和数学家才能找到语言来描述日益令人困惑的宇宙。阿里安罗德出色地讲述了数学革命背后的故事，这是推动十九世纪和二十世纪物理革命的引擎；这本书非常值得一读。”

‘There have been lots of books about the evolution of modern physics: from Newton to Maxwell to Einstein and on to quantum theory. But seldom does an author pay attention to the mathematical revolutions that made those physical theories possible. Only as the mathematical toolkit expanded from simple scalars to include such tools and ideas as quaternions and vectors and tensors could physicists and mathematicians find the language to describe an increasingly bewildering universe. Arianrhod does a remarkable job telling the story of the mathematical revolution under the hood, the engine that drove the physics revolutions of the nineteenth and twentieth centuries; the result is a book well worth your time.’

—查尔斯·赛菲 (Charles Seife)， 《零：一个危险想法的传记》作者

—Charles Seife, author of Zero: The Biography of a Dangerous Idea

‘在《向量》中，阿丽安罗德以优美的笔触向我们展示了数学并不是只有极客才居住的陌生世界。它是我们周围的世界。”

‘In Vector, Arianrhod shows, with beautiful ease, that maths is not some foreign world only geeks inhabit. It is the world around us.’

―亚当·斯宾塞，《亚当·斯宾塞数字大全》作者：你想知道的关于 1 到 100 的所有数字

―Adam Spencer, author of Adam Spencer's Big Book of Numbers: Everything you wanted to know about the numbers 1 to 100

“如果所有数学都消失了，”物理学家理查德·费曼认为，“这将使物理学倒退整整一周。”数学家马克·卡克反驳道：“倒退整整一周就是上帝创造世界的那一周。”阿丽安罗德说服我们，向量和张量就是上帝创造的产物之一。学生和老师应该一起阅读这本优秀的书。'

‘“If all mathematics disappeared,” physicist Richard Feynman opined, “it would set physics back precisely one week.” To which mathematician Mark Kac retorted, “Precisely the week in which God created the world.” Arianrhod persuades us that vectors and tensors are among those creations. Students and teachers should read this excellent book together.’

—Marjorie Senechal， 《数学情报员》主编

—Marjorie Senechal, editor-in-chief of The Mathematical Intelligencer

“阿丽安罗德善于将复杂的事情简单化，并且善于讲故事，在传达数学之美和力量方面无与伦比。威廉·罗文·汉密尔顿、詹姆斯·克拉克·麦克斯韦和阿尔伯特·爱因斯坦在这个关于一个改变我们世界的简单想法的戏剧性故事中栩栩如生。”

‘With a flair for exposition that makes the complex simple, and a gift for storytelling, Arianrhod is without peer in conveying the beauty, and power, of mathematics. William Rowan Hamilton, James Clerk Maxwell, and Albert Einstein come alive in this dramatic tale of a simple idea that changed our world.’

—阿米尔·亚历山大，《无穷小：一个危险的数学理论如何塑造了现代世界》一书的作者

—Amir Alexander, author of Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

“每个人都明白以特定速度朝特定方向移动意味着什么。但花了很长时间才开始用一个明确的概念——矢量来思考这种行为。Arianrhod 生动而详细的编年史解释了为什么矢量和张量是我们思考宇宙的最佳方式的核心。”

‘Everyone understands what it means to move at some particular speed in some particular direction. But it took a long time to start thinking of such behavior in terms of a single clarifying concept, the vector. Arianrhod’s lively and detailed chronicle explains why vectors and tensors are at the heart of our best ways to think about the universe.’

—肖恩·卡罗尔，《宇宙中最伟大的想法》一书的作者

—Sean Carroll, author of The Biggest Ideas in the Universe

关于空间、时间和数学变换的惊人故事

A surprising story of space, time, and mathematical transformation

罗宾阿里安罗德

ROBYN ARIANRHOD

新南威尔士大学出版社承认贝德加尔人是新南威尔士大学兰德威克校区和肯辛顿校区所在地未割让领土的传统所有者，并承认他们与国家和文化的持续联系。我们向过去和现在的贝德加尔长者致敬。

UNSW Press acknowledges the Bedegal people, the Traditional Owners of the unceded territory on which the Randwick and Kensington campuses of UNSW are situated, and recognises their continuing connection to Country and culture. We pay our respects to Bedegal Elders past and present.

新南威尔士大学出版社出版的书

A UNSW Press book

发布者

Published by

新南方出版公司

NewSouth Publishing

新南威尔士大学出版社有限公司

University of New South Wales Press Ltd

新南威尔士大学

University of New South Wales

悉尼新南威尔士州 2052

Sydney NSW 2052

澳大利亚

AUSTRALIA

https://unsw.press/

https://unsw.press/

© 2024 Robyn Arianrhod。保留所有权利。

© 2024 by Robyn Arianrhod. All rights reserved.

首次由芝加哥大学出版社在美国出版

First published in the United States by The University of Chicago Press

本书受版权保护。除《版权法》允许的出于私人学习、研究、批评或评论目的的合理使用外，未经书面许可，不得以任何方式复制本书的任何部分。如有疑问，请联系出版商。

This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review, as permitted under the Copyright Act, no part of this book may be reproduced by any process without written permission. Inquiries should be addressed to the publisher.

设计Lauren Michelle Smith

Design Lauren Michelle Smith

封面图片Jim.belk，Wikimedia Commons

Cover image Jim.belk, Wikimedia Commons

本书已尽一切合理努力获得版权资料的使用许可，但在某些情况下无法追踪版权。作者欢迎提供这方面的信息。

All reasonable efforts were taken to obtain permission to use copyright material reproduced in this book, but in some cases copyright could not be traced. The author welcomes information in this regard.

序幕

Prologue

（1）代数的解放

(1)The Liberation of Algebra

（2）微积分的到来

(2)The Arrival of Calculus

（3）向量的创意

(3)Ideas for Vectors

（4）了解空间（和存储）

(4)Understanding Space (and Storage)

（5）令人意外的新玩家和非常缓慢的接受度

(5)A Surprising New Player and a Very Slow Reception

（6）泰特和麦克斯韦：孵化电磁矢量场

(6)Tait and Maxwell: Hatching the Electromagnetic Vector Field

（7）从四元数到向量的缓慢旅程

(7)The Slow Journey from Quaternions to Vectors

（8）向量分析的最终成果——以及四元数的“战争”

(8)Vector Analysis at Last—and a “War” over Quaternions

（9）从空间到时空：向量的新转折

(9)From Space to Space-Time: A New Twist for Vectors

（10）弯曲空间和不变距离：通往张量的途径

(10)Curving Spaces and Invariant Distances: On the Way to Tensors

（11）张量的发明及其重要性

(11)Inventing Tensors—and Why They Matter

（12）一切汇聚在一起：张量和广义相对论

(12)Everything Comes Together: Tensors and the General Theory of Relativity

（13）接下来发生了什么

(13)What Happened Next

结语

Epilogue

时间线

Timeline

致谢

Acknowledgments

笔记

Notes

指数

Index

时不时地，我们对世界的理解就会出现惊人的突破。例如，颠覆了我们对太阳系中心位置的信念，相对论革命改变了我们看待时间和空间的方式，也改变了我们看待自己在宇宙中的位置的方式。电磁无线电波的惊人发现带来了无线技术，并对我们的日常生活产生了奇妙的改变，还有量子革命，它带来了看似神奇的新微技术，并以超自然的方式打破了我们对“现实”的观念。当然，还有数字革命，它仍在改变我们彼此交流的方式，尤其是在人工智能变得如此复杂的今天。这些突破带来了新的技术和文化时代，关于它们的文章也很多。然而，鲜为人知的是，这些科学和技术范式转变与同样引人注目的数学革命齐头并进。这是一个关于一些无名革命的故事：它们背后令人着迷的想法，以及使它们成为可能的人。

Every now and then, there’s a spectacular breakthrough in our understanding of the world. The upending of the belief in our central place in the solar system, for example, and the relativistic revolution that changed the way we see time and space—and the way we see our place in the cosmos itself. The dramatic discovery of electromagnetic radio waves, which led to wireless technology and its wondrous transformation of our everyday lives—and the quantum revolution, with its seemingly magical new microtechnologies and its preternatural shattering of our notions of “reality.” And then, of course, there’s the digital revolution that is still changing the way we communicate with each other, especially now that AI has become so sophisticated. These breakthroughs have brought into being new technological and cultural eras, and much has been written about them. It’s less well known, however, that these scientific and technological paradigm shifts went hand in hand with equally dramatic mathematical revolutions. This is a story about some of those unsung revolutions: the fascinating ideas behind them, and the people that made them possible.

从本质上讲，我在这里要讲述的故事是关于人类记录和理解我们周围所有数据的方式的演变。特别是，我将探索戏剧性的数学变换，这些变换为我们提供了称为“向量”和“张量”的非凡概念，因为它们是很多现代科学，以及我们的很多技术，都离不开这些语言。这些语言帮助我们揭开了宇宙的奥秘，就像我们是神一样。

At its core, the story I’ll tell here is about the evolution of the way we humans record and make sense of all the data that swirl around us. In particular, I’ll explore the dramatic mathematical transformations that gave us the remarkable concepts called “vectors” and “tensors,” for they underlie so much of modern science—and much of our technology, too. They are languages that have helped us uncover mysteries of the universe as if we were gods.

向量和张量具有如此强大威力的原因之一是，我们能够以一种全新且透明的方式处理空间维度，而这反过来又使我们能够发现新的自然法则以及这些法则的新技术应用。任何时候，只要你想利用空间中的位置，就需要处理这些维度，比如旋转机械臂、设计桥梁或风力涡轮机；弄清楚电动机或发电机中电磁力的影响，或者预测电磁波、水波甚至引力波的路径；绘制卫星轨迹或校准 GPS 等导航系统；或者你需要在空间或时空中进行分析的几乎任何事情。

One reason for this awesome power is that vectors and tensors made it possible to handle the dimensions of space in a new and transparent way— and this, in turn, made it possible to discover new laws of nature, and new technological applications of these laws. Anytime you want to utilise locations in space you need to handle these dimensions—rotating a robot arm, say, or designing a bridge or wind turbine; figuring out the effect of an electromagnetic force in a motor or generator, say, or predicting the path of an electromagnetic wave, a water wave, or even a gravitational wave; plotting the trajectory of a satellite or calibrating a guidance system such as GPS; or just about anything you need to analyse in space or space-time.

随着故事的展开，我们将更详细地了解向量和张量的物理威力。但这些语言不仅仅涉及物理维度，它们还涉及信息的“维度”。你可能读过关于“大数据”和信息革命的文章，但正是向量和张量帮助使数据变得可用和可理解——就像化学元素周期表既是化学中的组织工具又是理论工具一样，只不过我们故事中的数学应用范围要广泛得多。

We’ll see in more detail the physical power of vectors and tensors as the story unfolds. But these languages are not just about physical dimensions— they’re about “dimensions” of information, too. You’ve likely read about “big data” and the information revolution, but it is vectors and tensors that help make data usable and comprehensible—the way the periodic table of chemical elements is both an organisational and theoretical tool in chemistry, except that the math in our story is so much more widely applicable.

然而，向量和张量本身非常简单——至少从表面上看——因为你确实可以简单地将它们视为表示信息的简洁方式。例如，你可能还记得在学校学过，向量可以编码有关物理量（例如速度或力）的大小和方向的信息。因此，你可以用指向所需方向的箭头来表示它，而箭头的长度给出大小或“幅度”。张量增加了更多信息层，因此它们更像是多维数组而不是箭头。但是，当数学家发现这些箭头和数组如何相互组合的规则时，他们意识到他们已经找到了一种思考全新思想的全新语言。这是一个相当奇妙的想法。

Yet vectors and tensors themselves are remarkably simple—on the face of it, at least—for you can, indeed, begin by thinking of them simply as neat ways of representing information. For instance, you might remember from school that a vector can encode information about both the size and the direction of a physical quantity—say, a velocity or a force. So, you can represent it with an arrow pointing in the required direction, while the arrow’s length gives the size or “magnitude.” Tensors add in more layers of information, so they are like multidimensional arrays rather than arrows. But when mathematicians discovered the rules for how these arrows and arrays combine with each other, they realised they had found a brand-new language for thinking brand-new thoughts. And this is a rather wonderful idea.

一个简单的例子就是，几千年来，数学家只与数字打交道。实数的演化数字系统已经足够引人注目了，但这些数字只表达一件事：数量——重量、高度、距离、金额、苹果数量等等。另一方面，向量和张量可以同时编码多个事物，这就是为什么它们是如此出色的表示大量数据的方式。这些额外的信息意味着，与单个数字相比，向量和张量可以提供更丰富的工业或 IT 问题图景，或者物理模型。

A simple illustration of what I mean by this is that for thousands of years mathematicians worked only with numbers. The evolution of real number systems was remarkable enough, but these numbers express only one thing: quantity—the magnitude of a weight, height, distance, amount of money, number of apples, and so on. Vectors and tensors, on the other hand, encode several things at once, which is why they are such great ways of representing a lot of data. And this extra information means that vectors and tensors can offer a far richer picture of an industrial or IT problem, say, or a physical model, than a single number ever could.

第一个认识到向量语言威力的著名物理学家是十九世纪略显古怪的苏格兰领主詹姆斯·克拉克·麦克斯韦。我们稍后会正式介绍他，但他的电磁理论是第一个现代场论，他利用该理论破解了光的本质这一长期谜题，并预测了无线电波的存在——这一切都是一举成功。他最初的理论直观上是“向量的”，但当他了解到向量实际上是“一个东西”，有自己的数学规则时，他意识到它们是更简洁、更优雅地表达他的发现的正确工具。

The first major physicist to recognise the power of vector language was the gently eccentric nineteenth-century Scottish laird James Clerk Maxwell. We’ll meet him properly later, but his theory of electromagnetism was the very first modern field theory, which he used to crack the longstanding riddle of the nature of light and to predict the existence of radio waves—all in one fell swoop. His initial theory was intuitively “vectorial,” but once he learned that vectors were actually “a thing,” with their own mathematical rules, he realised they were the right tools for expressing his discovery more succinctly and elegantly.

起初，并没有多少人认真对待他：正如我们将看到的，他突破性地应用“矢量场”来表示自然界的电磁场……对于主流物理学家来说，这实在是太数学化、太“非物理”了。但是，如果麦克斯韦凭借其对矢量的出色应用获得认可已经如此困难，那么想象一下这种数学语言的创造者一定拥有的热情和自信。

Not that many people took him seriously at first: as we’ll see, his breakthrough application of “vector fields” to represent nature’s electromagnetic fields was … well, just too mathematical, too “unphysical,” for mainstream physicists. But if it was difficult enough for Maxwell to achieve recognition for his superb application of vectors, imagine the passion and self-belief the creators of this mathematical language must have had.

这个故事中的众多明星之一是爱尔兰数学家威廉·罗文·汉密尔顿。他是“向量”一词的创造者，也是第一个提出向量数学理论的人——他马上就意识到，他创造了一个如此新颖的东西，打破了数学家们几千年来理所当然的规则。然而，当他瞥见这种新语言的可能应用时，在麦克斯韦发表他的奇妙理论的六年前，他欣喜若狂。他欣喜地写信给一位与他一起研究这个问题的同事，“还有什么比这更简单、更令人满意的吗？你不觉得，也不认为，我们走在正确的道路上，以后会得到感谢吗？别介意什么时候……” ¹汉密尔顿不仅在说

One of the many stars of this story is the Irish mathematician William Rowan Hamilton. He’s the one who coined the term “vector” and first presented its mathematical theory—and he knew right away that he’d created something so new it broke a rule mathematicians had taken for granted for thousands of years. Yet when he glimpsed the possible applications of this new language, six years before Maxwell published his marvelous theory, he was over the moon. He wrote joyously to a colleague who was working with him on the subject, “Could anything be simpler or more satisfactory? Do you not feel, as well as think, that we are on a right track, and shall be thanked hereafter? Never mind when....”¹ Hamilton was speaking not only

这里是关于向量的，还有他发明的四元数，四维的包含向量的“数字”。正如我们所见，四元数可以完成向量所能做的一切，但它们在航天器制导和图像处理等特定任务的编程中效率更高，这只是两个现代应用。

about vectors here, but also his invention of quaternions, four-dimensional “numbers” that contain vectors. As we’ll see, quaternions can do everything that vectors can do, but they are more efficient in programming certain tasks in spacecraft guidance and image processing, to mention just two modern applications.

可怜的汉密尔顿从未得到足够的感谢：他于 1865 年去世，就在麦克斯韦发表电磁学理论几个月后，但他还没能将其改写成完整的矢量语言^。2

Poor Hamilton never did receive sufficient thanks: he died in 1865, just a few months after Maxwell published his theory of electromagnetism but before he’d been able to recast it into full vector language.²

至于张量，麦克斯韦在张量被发明之前就去世了，但我敢打赌他肯定也意识到了张量的威力。麦克斯韦去世的那一年正是爱因斯坦出生的那一年，这尤其具有象征意义——不仅因为爱因斯坦的理论受到了麦克斯韦理论的启发，还因为爱因斯坦对张量所做的贡献与麦克斯韦对矢量所做的贡献一样：他是第一位展示张量实际威力的物理学家。张量使他能够创造弯曲的时空和现代宇宙学学科——也使他能够预测引力波和透镜的存在，并准确量化引力对时间的影响，而现在正是利用这种影响，让 GPS 导航变得如此精确。

As for tensors, Maxwell died before they were developed, but I’m betting he would have recognised their power, too. He died the same year that Einstein was born, which is especially symbolic—not only because Einstein’s theories were inspired by Maxwell’s, but also because Einstein did for tensors what Maxwell had done for vectors: he was the first major physicist to show their practical power. They enabled him to create curved space-times, and the discipline of modern cosmology—and they enabled him to predict the existence of gravitational waves and lenses, and to accurately quantify the gravitational effect on time that is now used to make GPS directions so accurate.

实验物理学家花了四分之一世纪的时间在实验室中验证了麦克斯韦对无线电波的预测，而探测爱因斯坦的引力波则花了一百年的时间。这表明这些基于矢量和张量的理论是多么领先于时代。这种预测能力是数学语言令人兴奋的地方之一。就好像用数学描述物理现实的行为创造了一个放大镜，通过数学模式揭示了长期隐藏的潜在物理属性。我们将在后面看到具体的例子，但在这里我只想补充一点，量子理论也很好地利用了矢量和张量——到目前为止，它的任何预测都没有被推翻。

It took experimental physicists a quarter of a century to verify in the lab Maxwell’s prediction of radio waves, and it took a hundred years to detect Einstein’s gravitational waves. That’s an indication of how far ahead of the game these vector- and tensor-based theories were. This kind of ability to make predictions is one of the exciting things about mathematical language. It’s as if the act of describing physical reality mathematically creates a magnifying glass, revealing, through mathematical patterns, underlying physical attributes that had long lain hidden. We’ll see specific examples of this as we go, but here I’ll just add that quantum theory also makes fine use of vectors and tensors—and, so far, none of its predictions have been disproved.

由于向量和张量是存储和使用信息的方式，它们的用途当然远不止在物理学中。正如我之前所暗示的，它们在越来越多需要处理大量数据的领域中发挥着重要作用——从工程和遗传学到搜索引擎和人工智能，以及介于两者之间的更多领域。

Since vectors and tensors are ways of storing and using information, they’re useful far more widely than in physics alone, of course. As I intimated earlier, they are playing a fundamental role in a growing number of areas that need to handle a lot of data—from engineering and genetics to search engines and artificial intelligence, with much more in between.

然而，这些数学思想的全部威力的发展是如此惊人，影响如此深远，以至于我将它们的发现视为革命。将张量视为向量的概括是有帮助的，但这只是后见之明：从一种刚刚起步的向量语言发展到一种以严格方式包含向量和张量的复杂语言，花了三百年时间。而要获得向量概念的第一个新生迹象，已经花费了许多世纪——事实上，如果我们追溯到现存最古老的数学记录，则花费了数千年时间。因为向量和张量的历史与数据符号表示的历史息息相关，这些古代文献表明，找到表示信息的方法是数学故事的核心。

Yet the development of the full power of these mathematical ideas was so astonishing, and so far-reaching, that I’m treating their discoveries as revolutions. It’s helpful to think of tensors as a generalisation of vectors, but that is hindsight: it took three hundred years to move from a fledgling form of vector language to a sophisticated language that incorporates vectors and tensors in a rigorous way. And to get to that first, nascent hint of the vector concept, it had already taken many centuries—millennia, in fact, if we go back to the oldest surviving mathematical records. For the history of vectors and tensors is linked with the history of the symbolic representation of data, and these ancient documents show that finding ways to represent information is at the heart of the story of mathematics itself.

因此，我将从简要回顾这一切的起源开始讲述这个故事。当然，我在这里以及在后面章节中对这段漫长历史的讲述不可能详尽无遗。它必然是有选择性和主观性的。我的目标之一只是展示复杂的数学思想需要多长时间以及需要多少跨文化合作才能发展起来。这是一条漫长而曲折的道路，通往如今广泛使用的现代向量和张量分析，我想讲述的故事是一段思想之旅——这些思想往往令人惊讶，有时平淡无奇，但从一开始，它们就一直充满着可能性。

So, I’ll begin this tale with a brief tour back to the beginning of it all. Of course, my telling of this long history, here and as the story unfolds in later chapters, cannot be exhaustive. It’s necessarily selective and subjective. One of my goals is simply to show just how long it takes—and how much intercultural cooperation is needed—for sophisticated mathematical ideas to develop. It was a long and winding road to the modern vector and tensor analyses that are so widely used today, and the story I want to tell is a journey of ideas—ideas that are often surprising, sometimes mundane, but which have always been, right from the beginning, rich with possibility.

不过，如果在阅读本书的任何时候，你发现自己想浏览一下细节并继续阅读故事，那也没问题！

Still, if at any point throughout the book you find you want to skim the details and get on with the story, that’s fine, too!

楔形文字数学的发明者生活在底格里斯河和幼发拉底河之间的肥沃平原上——一千年后，希腊人将这片地区称为美索不达米亚（意为“两河之间”）。这里孕育了许多相互关联的文化，因此“美索不达米亚”一词至今仍被广泛用于描述在这个农业发达、知识丰富的地区发展起来的古代数学和其他文化创新。当然，这里并不是唯一一个从简单计数转向复杂数值算术的地方。但我们对美索不达米亚数学的了解远远超过对其他早期文化的了解，这仅仅是因为许多非凡的粘土文献幸存了下来。

The inventors of this cuneiform mathematics lived on the fertile plains between the Tigris and Euphrates Rivers—an area that the Greeks, a thousand years later, would call Mesopotamia (or “between two rivers”). It hosted a number of linked cultures, so the term “Mesopotamian” is still generally used to describe the ancient mathematical and other cultural innovations developed in this agriculturally and intellectually rich region. Of course, it was not the only place to move from simple counting to sophisticated numerical arithmetic. But we know so much more about Mesopotamian mathematics than that of other early cultures, simply because so many of those remarkable clay documents survived.

一些最早的、更为复杂的数字和数学泥板，可以追溯到近四千年前，上面写着乘法表。这些东西存在了这么久似乎令人惊讶，但当然，你需要能够进行加法和乘法才能完成基本的经济任务。历史学家已经洞察了这些早期任务的性质，因为这些功能也记录在泥板上。值得注意的是，其中一些最古老的文献包含表格，列出了各种正方形或矩形区域的边长，每个边长都与其面积相匹配——这种表格布局后来会演变成数学矩阵。简单的信息列表会演变成向量——但稍后我们会更多地讨论所有这些数学变化，因为它将向我们展示数学家如何从简单的会计列表转变为建模诸如电磁波或量子计算机所依赖的量子比特等复杂事物。与此同时，这些古老的表格对于计算潜在的粮食产量、种子需求、耕种田地所需的劳动力数量以及需要支付的工资和税款至关重要——这些都是任何大型社会为生产和分配食物和其他必需品所需要的东西^。3

Some of the earliest of the more sophisticated numerical and mathematical tablets, dating from nearly four thousand years ago, contain multiplication tables. It might seem surprising that such things have been around for so long, but of course you need to be able to add and multiply to carry out basic economic tasks. Historians have gained insight into the nature of those early tasks, because these functions, too, were recorded on clay tablets. Tellingly, some of the oldest of these documents contain tables giving lists of the lengths of the sides of various square or rectangular fields, each matched with their areas—the kind of tabular layout that would later morph into mathematical matrices. Simple lists of information would morph into vectors—but more on all this mathematical morphing later, for it will show us how mathematicians went from simple accounting lists to modeling such complex things as electromagnetic waves, for example, or the qubits that underpin quantum computers. Meantime, these ancient tables were vital for working out potential grain harvests, seed requirements, the amount of labour needed to work the fields, and the wages and taxes to be paid—the kinds of things that any large society needs in order to create and distribute food and other necessities.³

在最早的美索不达米亚先进社会中，经济正常运转所需的土地面积估计并不需要精确。测量员可以用木桩和绳子划出田地，然后测量其边长，但他们不必担心土地是否完全呈矩形，因为无论如何，土地都归国家所有。然而，大约公元前 1900 年，情况开始发生变化，普通民众也很快拥有了土地——这意味着测量需要变得更加准确，因为土地纠纷即将开始其漫长的历史。（相比之下，许多原住民一直沿用旧方法，直到殖民者打破了平衡。）因此，在最早的乘法表帮助会计师计算近似正方形和矩形面积的数百年后，美索不达米亚的测量员已经研究出如何制作完美的 90° 角——这表明他们可能在毕达哥拉斯之前一千年就发现了“毕达哥拉斯定理”。

In the earliest of these advanced Mesopotamian societies, the estimates of field sizes needed for the economy to run properly didn’t need to be exact. Surveyors could mark out fields with pegs and ropes, and then measure their sides, but they didn’t have to worry about making the land parcels exactly rectangular because the state owned it all anyway. From about 1900 BCE, however, things began to change, and soon ordinary folk could own land, too—and this meant that surveying needed to become more accurate, because land disputes were soon to begin their long history. (By contrast, many First Nations people practiced the old way right up until colonisation disrupted the balance.) And so, hundreds of years after the earliest multiplication tables had helped accountants calculate the areas of approximate squares and rectangles, Mesopotamian surveyors worked out how to make perfect 90° corners—which suggests they may have discovered “Pythagoras’s theorem” a millennium before Pythagoras.

一代又一代的小学生都在念诵这条古老的规则：“斜边的平方等于两邻边的平方和”。毕达哥拉斯生活在公元前六世纪，但大约 3700 年前的楔形文字泥板（最著名的是标有 Plimpton 322 和 Si.427 的泥板）包含了令人信服的证据，证明这条规则比它的希腊同名规则古老得多。Plimpton 322 泥板（图 0.1）已经破损，但残存的碎片列出了十五对与直角三角形的对角线和短边有关的数字。所使用的数字的选择以及表格各列的标题表明，这可能是一个隐含的、六十进制的“毕达哥拉斯三元组”列表，正如它们现在的名称一样，它是一组测量员可以从中选择的整数三元组。例如，(3, 4, 5) 是毕达哥拉斯三元组，因为 3 ² + 4 ² = 5 ²。石板 Si.427 支持这种解释，因为它是将一块土地私人细分为矩形和三角形的平面图——每个都有完美的矩形角和尺寸，符合毕达哥拉斯定理^。4

Generations of schoolchildren have chanted this ancient rule: “the square of the hypotenuse equals the sum of the squares of the two adjacent sides.” Pythagoras lived in the sixth century BCE, but cuneiform tablets dating from around 3,700 years ago—most famously those labeled Plimpton 322 and Si.427—contain compelling evidence that the rule is much older than its Greek namesake. The Plimpton 322 tablet (fig. 0.1) has been broken, but the surviving fragment lists fifteen pairs of numbers relating to the diagonal and the shorter side of a right-angled triangle. The choice of numbers used, and the headings in the table’s columns, suggest that this was probably a list of implicit, sexagesimal “Pythagorean triples,” as they are now called—a set of integer (whole number) triples that surveyors could choose from. For example, (3, 4, 5) is a Pythagorean triple, because 3² + 4² = 5². The tablet Si.427 supports this interpretation, for it is a plan of the private subdivision of a land parcel into rectangular and triangular fields—each with perfectly rectangular corners and dimensions that fit Pythagoras’s theorem.⁴

一千年后，古希腊数学家开始对测量一系列角度感兴趣，而不仅仅是 90°，因为他们不仅想测量地球，还想测量星星。天文学和数学一样，都是最古老的科学。毕竟，广阔而耀眼的夜空是一件奇妙的事情。由于他们无法测量到星星的距离，这些古希腊天文学家发现，你可以通过测量它们的角度来精确定位它们——因此他们还发现了两个关键的东西。首先，三角学。这并不是说早期的文明没有某种形式的角度制表和“三角”计算——历史学家仍在争论和解释楔形文字。但现存最古老的明确三角表存在于克劳狄斯·托勒密 1850 年前的希腊数学天文学汇编《天文学大成》中。（如果你忘记了三角学的基本思想，你可以提前看第 3 章中的图 3.4。）其次，希腊人发展了用坐标表示空间位置的思想——这一杰出的创新与新兴的矢量思想有很大关系。

A thousand years later, ancient Greek-speaking mathematicians were interested in measuring a range of angles, not just 90°, because they wanted to survey not only the earth but the stars, too. Along with mathematics, astronomy is the oldest science. After all, a vast and dazzling night sky is a wondrous thing. Since they had no way of measuring the distance to the stars, these ancient Greek astronomers figured out that you could pinpoint them by measuring their angles—and so they also discovered two key things. First, trigonometry. Which is not to say that earlier cultures didn’t have some form of angular tabulation and “trigonometric” calculation, too— historians are still debating and interpreting cuneiform texts. But the oldest extant, explicit trig table survives in Claudius Ptolemy’s extraordinary 1,850-year-old compilation of Greek mathematical astronomy, Almagest. (If you’ve forgotten the basic idea of trigonometry, you can look ahead to fig. 3.4 in chap. 3.) Second, the Greeks developed the idea of representing positions in space with coordinates—a brilliant innovation that has a lot to do with the emerging idea of a vector.

不过，在我进一步讨论这个问题之前，我想先在接下来几段中介绍一下与我们的故事相关的其他古代数学文化。例如，讲希腊语的托勒密生活在罗马统治下的埃及亚历山大——这提醒我们，随着帝国的扩张和衰落，残酷的动乱往往会随之而来，也提醒我们，在数学史上，“希腊”指的是讲希腊语的数学家的作品，无论他们的种族、民族或居住地如何。这也提醒我们，著名的亚历山大女数学家希帕蒂娅在三个世纪后对《天文学大成》进行了评论。虽然我强调现代数学的多元文化历史，但我应该指出，《天文学大成》是一个阿拉伯语单词，意思是“最伟大的”，这个词长期以来一直是用来描述托勒密编纂的非正式形容词，以区别于当时的其他作品。这本真正“最伟大”的书之所以能流传至今，很大程度上要归功于中世纪的阿拉伯语译者和注释者，因此，它自那时起就以阿拉伯语名称为人所知，这是再合适不过的了。

Before I talk a little more about this, though, in the next couple of paragraphs I want to acknowledge the other ancient mathematical cultures that also connect with our story. For instance, Greek-speaking Ptolemy lived in Alexandria, Egypt, under Roman rule—a reminder of the brutal upheavals that tend to follow as empires expand and fall, and also of the fact that in math history “Greek” refers to the work of Greek-speaking mathematicians, no matter their race, ethnicity, or where they lived. A reminder, too, of the famous female Alexandrian mathematician Hypatia, who wrote a commentary on Almagest three centuries later. And while I’m emphasising modern mathematics’ multicultural history, I should note that Almagest is an Arabic word meaning “the greatest,” a term that had long been the informal adjective used to describe Ptolemy’s compilation, to distinguish it from others of the time. This truly “greatest” book survives today largely because of its medieval Arabic translators and annotators, so it is fitting that it has been known ever since by its Arabic name.

和美索不达米亚人一样，古埃及人早在五千年前就开始用象形文字来表示数字，他们也对测量土地很感兴趣。当然，希腊人并不是第一个测量天空的人：美索不达米亚人、埃及人和许多其他古代民族都曾绘制过月亮和星星的轨迹，以便他们能够设计日历、祭拜神灵、穿越陆地和海洋，并注意到重要的巧合——比如，每年黎明前，当明亮的天狼星升起时，尼罗河就会开始泛滥。

Like the Mesopotamians, ancient Egyptians had begun to represent numbers some five thousand years ago—in their case, in hieroglyphic symbols—and they, too, were interested in surveying the land. And of course the Greeks were not the first to survey the sky: the Mesopotamians, Egyptians, and many other ancient peoples had charted the paths of the moon and stars so that they could devise calendars, propitiate gods, navigate their way across land and sea, and notice important coincidences—such as the fact that each year the Nile would begin to flood when the bright star Sirius rose just before dawn.

埃及和美索不达米亚的天文记录似乎是算术记录而非三角学记录——它们记录的是天体事件之间的天数，而不是不同时间恒星和行星的角度位置。但这些记录非常准确，以至于后来讲希腊语的天文学家很乐意使用它们。当时的大多数文化都没有留下天文观测的书面记录，但有些仍然以故事的形式流传下来——例如托雷斯海峡岛民传说中的 Baidam（或 Beizam），天鲨。这种“鲨鱼”是希腊人称为“熊”的星座，又名大熊座或北斗七星，它在夜空中的位置告诉人们何时播种庄稼——以及何时远离大海，因为鲨鱼正在繁殖！然后还有巨石阵、纽格兰奇和其他古代天文学仪式建筑，包括澳大利亚瓦萨鲁恩县的 Wurdi Youang 石圈，它可能比中东和欧洲的同类石圈早数千年。中国人、印度人、玛雅人、印加人和许多其他人也发展出了令人印象深刻的天文学知识——2017 年，国际天文学联合会通过在新的恒星命名系统中纳入八十六个土著星名，承认了所有这些古代文化的贡献^。5

It seems that Egyptian and Mesopotamian astronomical records were arithmetical rather than trigonometrical—they charted the number of days between celestial events, rather than the angular positions of stars and planets at different times. But these records were so accurate that later Greek-speaking astronomers were happy to make use of them. Most cultures from this time did not leave written records of their astronomical observations, but some still survive in stories—such as the Torres Strait Islander tradition of Baidam (or Beizam), the celestial shark. This “shark” is the constellation the Greeks called “the bear,” aka Ursa Major or the Big Dipper, and its position in the night sky told the people when to plant crops—and when to stay out of the sea because the sharks were breeding! Then there’s Stonehenge, New Grange, and other ancient astronomically aligned ceremonial structures, including the Wurdi Youang stone circle in Wathaurong country, Australia, which is likely thousands of years older than its Middle Eastern and European counterparts. The Chinese, Indians, Mayans, Incas, and many other people, too, developed an impressive knowledge of astronomy—and in 2017, the International Astronomical Union acknowledged the contributions of all these ancient cultures by including eighty-six indigenous star names in a new star naming system.⁵

至于古代对数学的贡献，我们将在接下来的两章中看到更多。但现在让我回到坐标，以及它们与向量的关系。这让我想起了托勒密，他给我们留下了最早的复杂记录，不仅是数学天文学，还有使用坐标来表示空间位置的记录。我不能把所有的功劳都归给他，无论是他的天文学还是他的坐标——辉煌的想法需要很长时间，需要很多人的贡献才能实现。但托勒密非常善于给予他的前辈应有的认可——这对他们来说也很好，因为他们的大部分工作后来都因霉菌、害虫、事故、战争而丢失了，而且由于《天文学大成》非常成功，早期的作品被丢弃了。6亚历山大的^欧几里得是一个显著的例外，因为他著名的《几何原本》的内容完整地保存了下来。欧几里得在公元前 300 年左右撰写了这本书，比托勒密的《天文学大成》早了大约 450 年，它为我们提供了至今仍在使用的理解日常（平坦）“欧几里得”空间的规则。

As for ancient contributions to mathematics, we’ll see a little more of those in the next two chapters. But now let me return to coordinates, and how they relate to vectors. Which brings me back to Ptolemy, who left us the earliest sophisticated record not just of mathematical astronomy, but also of the use of coordinates to represent positions in space. I cannot give him all the credit, for his astronomy or for his coordinates—it takes a long time, and contributions from many people, for brilliant ideas to come to fruition. But Ptolemy was very good at giving his forerunners due recognition—which is just as well for them, since most of their work was later lost to mold, pests, accidents, war, and the fact that Almagest was so successful that the earlier works were discarded.⁶ Euclid of Alexandria is a notable exception, for the content of his famous Elements survives in full. Euclid composed this book around 300 BCE, some 450 years before Ptolemy’s Almagest, and it gave us the rules we still use for understanding everyday (flat) “Euclidean” space.

然而，就现存的记录而言，托勒密引入了纬度和经度的概念，我们今天用它来定位地球上的位置。在他的著作《地理学》中，他列出了数千个地理位置——在《天文学大成》中，他有一个包含一千多颗恒星的星表，这些恒星是通过纬度和经度的天体类似物定位的。所有这些坐标都是角度——为了测量它们，托勒密描述了如何用阴影棒和像量角器一样的刻度盘来找到太阳的角度，而为了定位其他恒星，他使用了浑天仪。（浑天仪是一个复杂的可移动环网络，形成一个骨架球体，包围着一个微小的地球；这些环代表了太阳的视年和视日轨道、观察者的地平线和子午线——它们与观察者的纬度和经度有关——以及一个标有度数刻度的环和另一个标有观测星空的孔。）

As far as the surviving records go, though, it is Ptolemy who introduced the idea of latitude and longitude that we use today for locating places on Earth. In his book Geography, he lists thousands of geographical locations—and in Almagest, he has a star catalogue of more than a thousand stars, located via celestial analogues of latitude and longitude. All these coordinates are angles—and to measure them, Ptolemy described how to find the sun’s angle with a shadow stick and a graduated dial like a protractor, while to locate the other stars he used an armillary sphere. (This is a complex network of movable rings forming a skeletal globe that encloses a tiny Earth; the rings represent such things as the apparent yearly and daily orbits of the sun, the observer’s horizon and meridian—which are related to the observer’s latitude and longitude—as well as a ring marked with a degree scale and another with sighting holes to the stars.)

又过了一千二百年，我们才逐渐在数学课堂上习惯使用方格纸上的坐标，特别是通过十四世纪法国数学家的工作。Nicole Oresme（尽管与欧几里得同时代的人阿波罗尼乌斯也曾略微领悟过这一思想）。三个世纪后，另一位法国人勒内·笛卡尔更接近这一思想，后来的数学家们效仿他，牢固地建立了人们熟悉的x - y “笛卡尔”坐标系，其轴线在中间相交，即网格的“原点”。现在，在这个简短的初步概述中，我们将着重介绍矢量的概念。

It took another twelve hundred years to get an inkling of the kind of coordinates on graph paper that we’re used to in math classes today— particularly through the work of the fourteenth-century French mathematician Nicole Oresme (although Euclid’s near-contemporary Apollonius of Perga had also glimpsed the idea). Three centuries later, another Frenchman, René Descartes, came closer still, and, following his lead, later mathematicians firmly established the familiar x-y “Cartesian” coordinates, whose axes meet in the middle, at the “origin” of the grid. And now we are homing in, in this short preliminary overview, on the idea of a vector.

因此，我想用本介绍的剩余部分来进一步介绍现代向量是什么，以及它们能做什么。不过，首先，我想重申我在前面几页中概述的向量悠久、多元文化史前史的重要性，因为本书的大部分内容将集中在发明微积分、向量和张量的现代欧洲人的工作上。今天，数学是一项真正普遍的事业，但在我们故事的大部分发生的十九世纪，英国（当时包括爱尔兰）特别适合成为科学强国。它拥有历史悠久的大学，以及越来越多的机械学院和工人学院，用于成人教育。 （1874 年，职业女性终于有了自己的学院，稍后我们会介绍一些在第一所女子大学学习的女性。）英国还拥有完善的银行系统，可以进行创业技术投资；拥有不断发展的铁路网络，可以促进交流和市场发展；拥有丰富的原材料——英国自己的煤炭、水和钢铁，以及从殖民地掠夺的材料。这些因素支撑了十八和十九世纪的工业革命，科学和技术齐头并进，相互促进——当然，不仅仅是在英国。欧洲也有同样发达的机构和科学界——尤其是我们的故事中的法国、意大利和德国。然而，正如我们所看到的，以及我们将在接下来的两章中看到的，这些现代科学界很大程度上归功于他们古代和中世纪的多元文化先驱。

So, I want to spend the rest of this introduction showing a little more about what modern vectors are, and what they can do. First, though, I want to reiterate the importance of vectors’ long, multicultural prehistory that I sketched in the preceding pages, because most of this book will focus on the work of the modern European men who invented calculus, vectors, and tensors. Math today is a truly universal endeavor, but in the nineteenth century when much of our story takes place, the United Kingdom (which then included Ireland) was especially well placed to be a scientific powerhouse. It had long-established universities along with a growing number of Mechanics’ Institutes and Working Men’s Colleges for adult education. (Working women finally got their own college in 1874, and later we’ll meet some of the women who studied at the first University Women’s Colleges.) It also had a well-established banking system that enabled entrepreneurial technological investment, it had a growing railway network that facilitated communication as well as markets, and it had abundant raw materials—its own coal, water, and steel, as well as materials plundered from its colonies. These factors underpinned the Industrial Revolution of the eighteenth and nineteenth centuries, in which science and technology went hand in hand, each fueling the other—and not only in Britain, of course. Europe had similarly well-developed institutions and scientific communities—especially, for our story, France, Italy, and Germany. As we’ve seen, though, and as we’ll see in the next two chapters, these modern scientific communities owed much to their ancient and medieval multicultural forebears.

因此，接下来的几页只是简要的回顾，也许会提醒您两个基本思想，您可能已经了解过，但没有意识到它们是如此具有开创性。我们稍后会更详细地介绍它们，但我在这里提到它们是为了为我们在旅途中需要的一些术语和概念打下基础——同时也让您了解为什么向量在今天如此重要。张量是向量的多维表亲，将在故事的后面出现，因为它们将建立在为向量开发的概念和符号体系之上——而向量之旅出乎意料地是分层的。但随着我们探索这些层次，我们也将建立支撑张量的思想。

So, the next couple of pages are just a quick look ahead, and perhaps a reminder of two basic ideas you may have learned but didn’t realise were so groundbreaking. We’ll meet them in more detail later, but I mention them here to lay tracks for some terminology and concepts we’ll need on the journey—and also to give a glimpse into why vectors are so important today. Tensors, vectors’ multidimensional cousins, will come much later in the story, for they will build on the concepts and symbolism developed for vectors—and the journey to vectors is unexpectedly layered. But as we explore these layers, we’ll also be building up ideas that underpin tensors, too.

第一个想法是符号。数学符号有时看起来很神秘，所以我在这本书中的目标之一就是与你分享一些概念上的突破，这些突破使得为新想法发明新符号成为必要。即使是那些熟悉的x和y也是经过了很长时间的酝酿，我们将在下一章中看到。我们不会在接下来的几章中开始介绍现代矢量符号的诞生——这表明这些想法的演变需要多长时间。不过现在，我只会将这个故事与你可能已经知道的内容联系起来。例如，位置向量通常表示为r，$\overrightarrow{r}$， 或者$\widehat{r}$即使你不熟悉矢量，你也可能记得看到过教科书将这样的符号与普通字母和数字混淆。这是因为粗体类型（或字母顶部的“帽子”或箭头或字母下方的波浪线）是当今表示矢量的常用方式，以将它们与表示数字、大小而没有方向的普通字母区分开来。因此，在这种情况下，没有粗体类型或帽子或其他标记的r仅表示距离的大小，而不表示方向；像这样的量被称为“标量”，而不是“矢量”。另一个例子该符号的其中一种表示方式是， v代表速度，其中包括方向和速率，v代表速率——速度的大小。

The first idea is notation. Mathematical symbols can sometimes seem mysterious, so one of my aims in this book is to share with you some of the conceptual breakthroughs that made it necessary to invent new symbols for new ideas. Even those familiar x’s and y’s were a long time coming, as we’ll see in the next chapter. We won’t get to the inauguration of modern vector symbolism for several chapters—which goes to show just how long it took for these ideas to evolve. For now, though, I’ll just lay tracks that connect this story with what you may already know. For example, position vectors are often denoted by r, $\overrightarrow{r}$, or $\widehat{r}$. Even if you’re not au fait with vectors, you might remember seeing textbooks mixing up signs like this with ordinary letters and numbers. That’s because boldface type—or “hats” or arrows on top of letters or squiggles underneath them—are common ways of denoting vectors today, to distinguish them from ordinary letters that represent numbers, magnitudes without direction. So, in this case r, with no bold type or hat or other marker, would represent just the magnitude of the distance, with no reference to the direction; quantities such as this are called “scalars,” as opposed to “vectors.” Another example of this notation is v for velocity, which includes direction as well as speed, and v for speed—the magnitude of the velocity.

第二个关键概念是矢量有“分量”。在书的一页或一张方格纸的二维空间中，有两个独立的空间方向（通常表示为x和y方向或轴），因此点有两个坐标，它们之间的箭头有两个分量。换句话说，分量是从每个轴测量箭头时得到的值，如图 0.2所示。

The second key notion is that a vector has “components.” In the twodimensional space of a page in a book, say, or a piece of graph paper, you have two independent spatial directions—commonly represented as the x and y directions or axes—so your points have two coordinates and the arrow between them has two components. In other words, components are the values you get for your arrow when you measure it from each axis, as you can see in figure 0.2.

同样，在三维空间中，有三个独立的方向和三个分量。然后是时空，其中有四个坐标轴，三个空间轴和一个时间轴，而你的隐喻箭头有四个分量——所以你的位置矢量将测量不仅要测量行进的距离，还要测量所用的时间。这可能比简单地测量三个空间分量并从手表上读取时间分量要复杂一些——因为如果你相对于要测量的对象移动，时间和空间坐标会以混合的方式发生变化。当你处于引力场中时，情况会更加复杂——这是广义相对论而非狭义相对论的领域。我稍后会更多地谈论相对论，但在这里我只想指出时空中的矢量有四个分量。 （我还应该指出，空间和时间的这种混合是为什么在现代物理学论文中“spacetime”的拼写可能比“space-time”更常见的原因，因为它表明空间和时间是相互交织的——它们不仅仅是两个不同的东西互相嫁接在一起。但这两种拼写在今天的数学和物理学教科书中都有使用，而连字符版是该概念的创始人赫尔曼·闵可夫斯基和爱因斯坦使用的。我们稍后会介绍闵可夫斯基和爱因斯坦。）

Similarly, in three-dimensional space, there are three separate directions and three components. Then there’s space-time, where you have four coordinate axes, three spatial ones and a time axis, and your metaphorical arrows have four components—so your position vector would measure not only the distance traveled but also the time it took. It can be a bit trickier than simply measuring the three spatial components and reading off the time component from your wristwatch—because time and space coordinates change, in an intermingled way, if you’re moving relative to what you want to measure. It’s even more complicated when you’re in a gravitational field—the province of general rather than special relativity. I’ll talk more about relativity later, but here I just want to note that vectors in spacetime have four components. (I should also note that this intermingling of space and time is why the spelling “spacetime” is perhaps more common than “space-time” in modern physics papers, because it shows that space and time are interwoven—they are not just two different things grafted on to each other. But both spellings are used in maths and physics texts today, and the hyphenated version is the one used by the founder of the concept, Hermann Minkowski, and by Einstein. We’ll meet Minkowski and Einstein later.)

这个故事中极具争议的一件事就是箭头和其组成部分之间的细微差别。当人们最终掌握了这个微妙之处时，不仅为现代本科矢量数学的创造打开了大门，而且也为张量的创造打开了大门。我们将看到，产生混淆的部分原因是您也可以通过列出它们的组成部分来表示矢量，这比画箭头更容易、更经济——尤其是当您想脱离三维空间时。例如，假设您以 35 英里/小时的速度向北行驶（沿着一条平坦的二维街道——数学家们喜欢从简化假设开始探索新想法！），并且您选择南北方向作为y轴。从几何上讲，您的速度矢量将在图 0.2中沿y轴向上指向，并且它的长度为 35 个单位。但您也可以将此速度写为v = (0, 35)，这意味着您沿y轴以 35 英里/小时的速度移动，沿x轴以 0 英里/小时的速度移动。因此，标量 35 表示您的速度（单位为英里/小时），矢量 (0, 35) 表示速度朝北 ( y ) 方向。（用矢量符号标记 (0, 35)，如v = (0, 35)，可将其与笛卡尔网格上的单个点区分开来 - 因此您可以看到符号的重要性。）您可以像这样表示任何速度，无论方向如何。例如，如果你以每小时 35 英里的速度向东北行驶，你的矢量分量将是$\left(\frac{\mathrm{三十五}}{\sqrt{2}}，\frac{\mathrm{三十五}}{\sqrt{2}}\right)$（正如我在本尾注⁷中所展示的）。

One of the things that will prove extremely controversial in this story is the subtle difference between the arrow, so to speak, and its components. When the subtlety is finally grasped, the way will open for the creation not just of modern undergraduate vector maths, but of tensors, too. We’ll see that the confusion arose partly because you can also represent vectors simply by listing their components, and this is easier and more economical than drawing an arrow—especially when you want to move out of three dimensions. For example, suppose you’re driving at 35 mph in a northerly direction (along a flat, two-dimensional street—mathematicians love to explore new ideas by starting with simplifying assumptions!), and you choose the north-south direction to be the y axis. Geometrically, your velocity vector would point upward along the y axis in figure 0.2, and it would be 35 units long. But you can also write this velocity as v = (0, 35), meaning you’re moving 35 mph along the y axis and 0 mph along the x axis. So, the scalar, 35, gives your speed (in units of mph), and the vector, (0, 35), shows that the speed is in the northerly (y) direction. (Labeling (0, 35) with a vector symbol, as in v = (0, 35), distinguishes it from a single point on a Cartesian grid—so you can see the importance of notation.) You can represent any velocity like this, no matter the direction. For example, if you were traveling, say, northeast at 35 mph, your vector components would be $\left(\frac{35}{\sqrt{2}},\frac{35}{\sqrt{2}}\right)$ (as I’ve shown in this endnote⁷).

更一般地，暂时跳过从数字到符号代数的漫长道路，你可以用（v _x，v _y，v _z）表示三维空间中的速度分量，这表示你在x方向以v _x英里每小时的速度行进，在y方向以v _y英里每小时的速度行进，在 z 方向以v _z英里每小时的速度行进。更一般地，三维空间中的速度分量可以用（v ₁，v ₂，v ₃）表示，因为笛卡尔x，y，z坐标系不是唯一可用的坐标系。例如，在二维中，极坐标（r，θ）可用于定位圆周上的点，如图0.3所示；矢量（v ₁，v ₂）表示你在r方向以v ₁英里每小时的速度行进，在 θ 方向以v ₂英里每小时的速度行进。要定位球面上的某个点，可以使用球面坐标 (r, θ, φ) 将其扩展到三维，其中 φ 是矢量与z轴之间的角度。但细节在这里并不那么重要：这种符号背后的故事才是如此有趣——当然还有应用。

More generally—and skipping blithely, for the moment, over the long road from numbers to symbolic algebra—you might represent velocity components in three-dimensional space by (v_x, v_y, v_z), indicating that you’re going v_x mph in the x direction, v_y mph in the y direction, and v_z mph in the z direction. Even more generally, the components of velocity in 3-D space can be denoted by (v₁, v₂, v₃), because the Cartesian x, y, z coordinate system is not the only one available. For instance, in two dimensions polar coordinates (r, θ) are useful for locating points on the circumference of a circle, as you can see in figure 0.3; the vector (v₁, v₂) would mean you’re traveling v₁ mph in the r direction, v₂ mph in the θ direction. To locate a point on a sphere, you can extend this to three dimensions using spherical coordinates (r, θ, φ), where φ is the angle between the vector and the z axis. But the details are not so important here: it’s the story behind this symbolism that is so interesting—and, of course, the applications.

例如，在四维时空中，爱因斯坦将他的坐标标记为（x ₁，x ₂，x ₃，x ₄），其中前三个是通常的空间坐标，第四个是时间坐标。因此，该系统中四维速度矢量（或“四速度” ⁸）的分量将被标记为（v ₁，v ₂，v ₃，v ₄）。

For example, in four-dimensional space-time, Einstein labeled his coordinates as (x₁, x₂, x₃, x₄), where the first three are the usual spatial coordinates and the fourth is the time coordinate. So, the components of a four-dimensional velocity vector (or “four-velocity”⁸) in this system would be labeled (v₁, v₂, v₃, v₄).

以此类推，我们可以想象任意数量的维度。将向量中的信息表示为一串分量的美妙之处就在于此：如果你可以写出（v ₁，v ₂，v ₃，v ₄），那么有什么能阻止你写出（v ₁，v ₂，v ₃，v ₄，v ₅，...）任意数量的（有限！）维度呢？这正是弦理论家在提出我们的宇宙可能不只有四个维度，而是十个、十一个甚至二十六个维度时所做的事——有几种不同的弦理论。你可以将额外的维度视为使空间能够容纳微小弦振动的多种方式，每种方式代表不同的基本粒子或力。

By analogy, it’s possible to imagine any number of dimensions. This is the beauty of representing the information in a vector as a string of components: if you can write (v₁, v₂, v₃, v₄), what’s to stop you writing (v₁, v₂, v₃, v₄, v₅, …) for as many (finite!) dimensions as you like? That’s just the kind of thing string theorists have done when they suggest our universe might have not four but ten or eleven or even twenty-six dimensions—there are several different string theories. You can think of the extra dimensions as enabling space to accommodate the many ways in which tiny strings can vibrate, each way representing a different fundamental particle or force.

弦理论是数学“向量空间”中的维度可以表示比普通空间和时间更奇特事物的一个例子。在量子力学中，电子的磁取向轴或“自旋”可以是“向上”或“向下”——或者是两者的“叠加”，就好像电子无法决定它希望自旋向上还是向下。它类似于二维速度矢量，其中一个分量在x方向上，另一个分量在y方向上，因此对于电子自旋，你可以让两个自旋方向“向上”和“向下”作为轴。

String theory is an example of the way the dimensions in a mathematical “vector space” can represent more exotic things than ordinary space and time. In quantum mechanics, to take another example, the axis of the magnetic orientation, or “spin,” of an electron can be “up” or “down”—or a “superposition” of the two, as if the electron can’t decide whether it wants its spin to be up or down. It’s analogous to a two-dimensional velocity vector having one component in the direction x, say, and another in the direction y, so for electron spin you can let the two spin directions “up” and “down” be your axes.

同样，在计算中，你用两个二进制数字（“位”）0 和 1 来编码信息，它们在物理上可以用例如打开和关闭电开关来表示。在量子计算中，位的类似物是量子比特（发音为“cue-bit”，是“量子比特”的缩写）。量子 0 和 1 可以在物理上表现为电子自旋的两个基本状态——0 可以用自旋“向上”表示，1 可以用自旋“向下”表示——因此，量子比特也可以在数学上表示为二维矢量。

Similarly, in computing you encode information with two binary digits (“bits”), 0 and 1, which are represented physically by, for example, turning an electric switch off and on. In quantum computing, the analogue of a bit is the qubit (pronounced “cue-bit,” short for “quantum bit”). Quantum 0’s and 1’s can be manifested physically as the two fundamental states of electron spin—0 might be represented by spin “up” and 1 by spin “down”—so qubits, too, can be represented mathematically as two-dimensional vectors.

向量在现代商业和技术应用中也很有用。例如，轴或“维度”可能代表网站问卷或政治调查中的不同问题，或影响房价或其他社会经济数据，或图像处理中像素的位置和颜色。顺便说一句，如果你熟悉图像处理，你就会知道在某些计算应用程序中，“矢量”一词的用法与我一直以来概述的通常数学含义略有不同。图像处理中的“矢量文件”使用方程式来生成形状，而不是使用矢量来存储有关像素等的信息。这些“矢量文件”中生成的形状确实沿着形状从一点“携带”到另一点的线，“载体”是“矢量”的本义。将数学矢量看作箭头，你可以看到它们也将线从箭头的尾部带到箭头的尖端。尽管如此，箭头，或者更一般地说是其组成部分的字符串，是大多数应用程序（物理、商业和计算）中使用的矢量类型。

Vectors are useful in modern business and tech applications, too. For instance, the axes or “dimensions” might represent different questions on a website questionnaire or political survey, or the different factors affecting house prices or other socioeconomic data, or the positions and colours of pixels in image processing. By the way, if you’re familiar with image processing, you’ll know that in some computing applications, the word “vector” can be used slightly differently from the usual mathematical meaning that I’ve been outlining. The “vector files” in image processing use equations to generate shapes, rather than vectors for storing information about such things as pixels. The shapes produced in these “vector files” do “carry” lines from point to point along the shape, and “carrier” is the original meaning of “vector.” Looking at mathematical vectors as arrows, you can see how they, too, carry the line from the tail to the tip of the arrow. Still, the arrow, or more generally the string of its components, is the kind of vector used in most applications—in physics, business, and computing.

但是威廉·罗文·汉密尔顿 (William Rowan Hamilton) 并没有因为可以写下任意数量的组成部分来表示各种物理、数字和经济系统或场景而感到兴奋。自从人类能够计数和书写以来，存储和表示数据一直很重要，但计算也是如此 - 向量和张量使您能够同时进行表示和计算。这就是它的魔力所在：尽管将向量写为组成部分列表简单且实用，但汉密尔顿如此兴奋的是，当你将向量视为一个整体，而不仅仅是其各个组成部分时，向量算术规则使向量比单纯的数字更强大。

But William Rowan Hamilton didn’t get so excited just because you can write down any number of components to represent various physical, digital, and economic systems or scenarios. Storing and representing data has been important ever since humans could count and write, but so has computation—and vectors and tensors enable you to do both the representing and the calculating at the same time. That’s where the magic comes into it: despite the simplicity and utility of writing vectors as lists of components, what Hamilton was so thrilled about was that when you consider the vector as a whole, not just its individual components, the rules of vector arithmetic make vectors far more powerful than numbers alone.

例如，将麦克斯韦方程写成全矢量形式，可以相对容易地推导出电磁波方程——麦克斯韦曾预言过电磁波的存在。磁场矢量和矢量积在数学上定义电子等亚原子粒子的自旋方面也发挥着作用（再举一个例子），从而使自旋能够应用于现实世界——例如，在磁共振成像（MRI）中，它用于通过“观察”患者体内来诊断患者。

For example, writing Maxwell’s equations in whole vector form makes it relatively easy to deduce the equations for electromagnetic waves—a phenomenon whose very existence Maxwell had predicted. Magnetic field vectors and vector products also play a role in mathematically defining the spin of subatomic particles such as electrons (to take just one more example), thereby enabling spin to be applied in the real world—in magnetic resonance imaging (MRI), say, which is used to diagnose patients by “seeing” inside their bodies.

汉密尔顿做梦也想不到这样的事情。但是，一旦他的矢量算法问世，它的模式就引起了其他数学家的兴趣——尤其是那些喜欢看看他们能把逻辑推向多远的人。数学规则和语法的后果。有时他们会走得太远，以至于陷入一种新的现实，就像爱丽丝穿过镜子掉进兔子洞一样。

Hamilton couldn’t have dreamed of such things. But once his vector arithmetic was out in the world, its patterns intrigued other mathematicians—especially those who like to see how far they can push the logical consequences of the rules and grammar of mathematics. And sometimes they push so far that they tumble into a new kind of reality, like Alice through the looking glass and down the rabbit hole.

汉密尔顿关于乘法的看似荒谬的发现，以及随后所有突破性的矢量和张量发展和应用，都需要使用代数符号。我说的是“矢量算术”，但更一般地说是“矢量代数”。简单地说，算术与数字有关，而代数​​与字母有关，字母代表数字或可量化的量，如温度或速度。这种符号使数学家能够概括。例如，用符号表示普通算术的基本规则（如交换律，a + b = b + a和a × b = b × a）可以让你看到适用于所有数字的模式，这样你就不必像最早的数学家那样单独列出每个例子。汉密尔顿尽可能遵循这些基本算术模式，从而发展了他的矢量代数规则，因此当他意识到必须拓宽乘法规则时，他和其他人一样感到震惊。

Hamilton’s seemingly absurd discovery about multiplication, and all the breakthrough vector and tensor developments and applications that followed, required the use of algebraic symbolism. I spoke of “vector arithmetic,” but more generally this is “vector algebra.” Speaking loosely, arithmetic works with numbers, and algebra works with letters that symbolise numbers or quantifiable quantities, such as temperature or speed. This symbolism enables mathematicians to generalise. For example, writing the basic rules of ordinary arithmetic symbolically—such as the commutative law, a + b = b + a and a × b = b × a—enables you to see patterns that hold for all numbers, so you don’t have to list each example separately the way the earliest mathematicians had to do. Hamilton had developed his rules of vector algebra by following, as best he could, these basic arithmetic patterns, so he was as shocked as anyone when he realised he had to broaden the rules for multiplication.

但符号代数思维的出现已经很久了——远远晚于埃及人和美索不达米亚人、古希腊人和中国人、中世纪阿拉伯人和印度人，以及所有其他数学17 世纪之前的文化。即使在 19 世纪，许多数学家也不信任纯粹抽象的符号代数，因为它似乎与日常经验脱节。因此，为了介绍汉密尔顿、麦克斯韦和其他矢量先驱将面临的挑战，下一章将从这一概述回到一个故事，讲述人们最初如何开始用字母表示和计算——以及这个看似简单的步骤如何使数学家能够更有创意地思考。

But symbolic algebraic thinking had been a long time coming—long after the Egyptians and Mesopotamians, the ancient Greeks and Chinese, the medieval Arabs and Indians, and all the other mathematical cultures before the seventeenth century. Even in the nineteenth century many mathematicians mistrusted purely abstract symbolic algebra, so unmoored from everyday experience did it seem. So, to set the scene for the challenges that Hamilton, Maxwell, and the other vector pioneers would face, the next chapter moves back from this overview to a story about how people began to represent and calculate with letters in the first place—and how this seemingly simple step has enabled mathematicians to think more creatively.

不过，首先，我们将站在汉密尔顿的角度，重温他疯狂而超越常理的那一刻：古老的数学规则终究是可以被打破的。

First, though, we’ll walk in Hamilton’s shoes for a mile or two, reliving the moment he had his wild, transgressive realisation that the ancient rules of mathematics could, after all, be broken.

每年 10 月 16 日，数学爱好者都会从邓辛克天文台出发，穿过田野，步行到皇家运河。运河有许多交叉口，但他们特别要前往其中一条：布鲁姆桥。他们正在庆祝汉密尔顿日——爱尔兰最伟大的数学家威廉·罗恩·汉密尔顿——他们正在重现数学史上最著名的一次散步。他们正在纪念 1843 年 10 月 16 日，当天，爱尔兰皇家天文学家汉密尔顿正前往主持爱尔兰皇家科学院的会议。

Every year on October 16, mathematically inclined Dubliners take a communal walk across the fields from Dunsink Observatory and down to the Royal Canal. The canal has many crossings, but they’re heading for one in particular: Broome Bridge. They are celebrating Hamilton Day—that’s William Rowan Hamilton, Ireland’s greatest mathematician—and they’re reenacting one of the most famous walks in the history of mathematics. They are commemorating October 16, 1843, when Hamilton—Ireland’s royal astronomer—was on his way to preside over a meeting of the Royal Irish Academy.

汉密尔顿的妻子海伦也加入了他的散步，但尽管环境很美，尽管他被浪漫主义所吸引，但这并不是传统意义上的浪漫散步：他太专注了。多年来，他一直在努力解决一个看似棘手的问题，这个问题一直困扰着他：如何在三维空间中用代数表示几何运算，比如旋转——操纵机器人、逼真的计算机图像或航天器所需的旋转，汉密尔顿并没有考虑这些高科技的可能性。他感兴趣的只是解决数学问题3-D 旋转，所以他仍在尝试发明最终有助于实现这些应用的数学。然后，当他经过布鲁姆桥时，他突然想到。你只能用 4-D 数学来做 3-D 旋转^。1

Hamilton’s wife, Helen, had joined him on his walk, but lovely as the setting was and much as he was drawn to Romanticism, this was not a romantic walk in any traditional sense of the word: he was much too preoccupied. He’d been wrestling for years with a seemingly intractable problem that simply would not leave him alone: how to represent algebraically geometrical operations such as rotations, in three-dimensional space—the kind of rotations you need to manipulate a robot, a realistic computer image, or a spacecraft, say, not that Hamilton was thinking about such high-tech possibilities. He was interested simply in solving the mathematical problem of 3-D rotations, so he was still trying to invent the maths that would eventually help make these applications possible. And then, as he passed by Broome Bridge, it hit him. You could only do 3-D rotations with 4-D maths.¹

这个见解已经足够吸引人了，但还不止于此。为了使他的四维数学成立，汉密尔顿必须改变乘法的基本规则。你不能再假设交换律a × b = b × a，这个律在普通数相乘时显然是正确的。你可能还记得与向量乘法的“右手定则”搏斗，但我们将在第 4 章中正确地了解它，因为它提供了一种简洁的方式来证明向量积不交换。然而，学校和大学的向量分析是事后才知道的。回到 1843 年，这种打破所谓交换律的行为是惊人的大胆，甚至是冒昧的：在什么样的宇宙中说 2 × 3 的某个类似物不等于 3 × 2 是有意义的？也许在刘易斯·卡罗尔的《仙境》中，疯帽子告诉爱丽丝，“说出你的意思”并不等同于“你想说的话”，因为“你也可以说‘我看到我吃的东西’和‘我吃我看到的东西’是一回事！” ²但这在数学家几千年来一直居住的直觉精神世界中肯定说不通。汉密尔顿非常兴奋，他拿出小刀，当场在石桥上刻下了他的魔法公式。

This was an intriguing enough insight, but there was more. To make his 4-D maths work, Hamilton had to change the fundamental rules of multiplication. No longer could you assume the commutative law a × b = b × a, which is so obviously true when you multiply ordinary numbers. You might remember wrestling with the “right-hand rule” for vector multiplication, but we’ll meet it properly in chapter 4, for it gives a neat way of showing that vector products aren’t commutative. School and college vector analysis is hindsight, however. Back in 1843, this breaking of the so-called commutative law was astoundingly daring, even presumptuous: In what universe did it make sense to say that some analogue of 2 × 3 does not equal 3 × 2? Perhaps in Lewis Carroll’s Wonderland, where the Mad Hatter tells Alice that “say what you mean” is not the same as “mean what you say,” for “you might just as well say that ‘I see what I eat’ is the same thing as ‘I eat what I see’!”² But it certainly did not make sense in the intuitive mental world mathematicians had inhabited for several thousand years. Hamilton was so excited that he took out his penknife and scratched his magic formula right then and there on the stone bridge.

这幅传奇数学涂鸦的创作地，是现代人在汉密尔顿日的目的地。原作早已被风雨侵蚀，但我们从汉密尔顿的信件中知道它曾经存在过——在过去的半个世纪里，一块牌匾标记着他灵感迸发的地点，正如他后来回忆的那样，灵感就像电路闭合，火花迸发。2019 年，桥边人行道上的一幅纪念艺术作品以这个电气隐喻为基础，用电光照亮了这幅著名涂鸦的实质，

The site of this legendary piece of mathematical graffiti is the destination of those modern walkers on Hamilton Day. The original carving has long been lost to the wind and rain, but we know it existed from Hamilton’s letters—and for the past half century, a plaque has marked the spot where he experienced the inspiration that came to him, as he later recalled, as if an electric circuit closed and a spark flashed forth. A 2019 commemorative artwork in the pavement by the bridge built on this electrical metaphor, illuminating with electric light the substance of the famous graffiti,

i2 = j2 = ^k2 = ^ijk = −1 。^​

i² = j² = k² = ijk = −1.

如果汉密尔顿能看到的话！³

If only Hamilton could have seen it!³

你可能已经注意到，汉密尔顿公式中的i、j和k是“虚数”，因为没有“实数”的平方可以等于负数。（当你计算一个实数的平方时，你总是将两个同号的数相乘，所以你得到的平方总是正数。这意味着如果你得到的平方是负数，如i ² = −1，那么这个数i就不是实数，而是所谓的虚数。）

You might have already spotted that the i, j, and k in Hamilton’s formulae are “imaginary” numbers, for no “real” number squared can equal a negative number. (When you’re taking the square of a real number, you’re always multiplying two numbers of the same sign, so you always get a positive square. Which means that if you get a negative square, as in i² = −1, then such a number i is not a real number; rather, it’s a so-called imaginary one.)

稍后我将探讨汉密尔顿方程的含义，但目前，我想强调的是世界各地的朝圣者继续参观布鲁姆桥的原因。这组看似简单的方程包含了一种新型四维结构的关键，它开辟了一种全新的数学语言，如今已成为各种领域不可或缺的一部分。汉密尔顿将他的四维作品命名为“四元数”；它们包含两部分，一个一维实数，他称之为“标量”，以及一个具有大小和方向的三维（三分量）量，他称之为“矢量”。如果你熟悉现代矢量，那么出于实际目的，你可以用同样的方式思考汉密尔顿的矢量。（我们稍后会看到它们之间微妙的概念差异。它将在我们的故事中扮演一个有争议的角色。）

I’ll explore the meaning of Hamilton’s equations later, but for the moment, I want to emphasise the reason pilgrims from around the world continue to visit Broome Bridge. This seemingly simple line of equations contained the key to a new kind of four-dimensional construct, which opened up a whole new mathematical language that has become indispensable in all sorts of fields today. Hamilton gave the name “quaternions” to his 4-D creations; they contained two parts, a 1-D real number, which he called a “scalar,” and a 3-D (three-component) quantity having magnitude and direction, which he called a “vector.” If you’re familiar with modern vectors, then for practical purposes you can think of Hamilton’s vector in just the same way. (We’ll see the subtle conceptual difference between them later. It will play a controversial role in our story.)

在那次传奇般的行走之后的二十年，詹姆斯·克拉克·麦克斯韦 (James Clerk Maxwell) 创立了电磁场的矢量场理论。而更为复杂的四元数则等了将近 150 年才找到我之前开始列出的现实世界应用——机器人技术、CGI、分子动力学、手机屏幕的旋转、航天器控制等等。例如，月球漫步者尼尔·阿姆斯特朗 (Neil Armstrong) 就非常了解四元数在飞机和航天器导航中发挥的作用。他既是航空工程师，也是宇航员，在生命的最后阶段，他在访问都柏林期间向汉密尔顿表达了自己的敬意^。4

Two decades after that storied walk, James Clerk Maxwell created his vector field theory of electromagnetism. The more complicated quaternions had to wait nearly 150 years to find the real-world applications I began to list earlier—robotics, CGI, molecular dynamics, the rotations of our mobile phone screens, spacecraft control, and much more. Moonwalker Neil Armstrong, for one, knew all about the role quaternions now play in airplane and spacecraft navigation. He was an aeronautical engineer as well as an astronaut, and toward the end of his life he paid his own personal tribute to Hamilton during a visit to Dublin.⁴

在我们进一步了解汉密尔顿是如何成功完成他的探索的——以及四元数和向量能做什么之前——我想告诉你，他是如何设想出虚数和高维数学的。在序言中，我跳过了从三维到四维或更多维的转变，但这种思维转变远非一帆风顺。这是因为几千年来，数学基本上只有两种，算术和几何。算术是关于计算具体的数量，例如金钱、重量和距离——几何也是通过点、线、平面和形状以直观的方式呈现的，你可以在二维页面上绘制它们，也可以在三维空间中表示它们。如果我们想用物理模型模拟我们日常物理现实经验中可以轻易想象到的事物，那么三维就是我们能达到的极限。

Before we take a closer look at how Hamilton succeeded in his quest— and at what you can do with quaternions and vectors—I want to tell you how it came about that he could envisage such things as imaginary numbers and higher-dimensional maths. In the prologue, I skipped through the move from three dimensions to four or more, but this transition in thinking was far from straightforward. That’s because for thousands of years there had essentially been only two kinds of mathematics, arithmetic and geometry. Arithmetic was about counting concrete quantities, such as money, weight, and distance—and geometry, too, was visualised in a palpable way, through points, lines, planes, and shapes that you could draw on a two-dimensional page or represent in three-dimensional space. Three dimensions are as far as we can go if we want to physically model the kinds of things we can readily imagine with our everyday experience of physical reality.

那么，汉密尔顿的四维数学又如何呢？你如何进行这样的数学运算——它如何表示像在普通空间中旋转的物体这样具体的东西？这是一个很长的故事，因为数学家花了很长时间才学会抽象思维——将他们的想象力从有形事物中解放出来。要做到这一点，他们首先必须学会符号思维——从具体的算术和几何转向抽象的代数符号世界。所以，如果你曾经为代数而苦恼，请振作起来：这段悠久的历史表明，一代又一代的著名数学家也曾这样做过。

So, what of Hamilton’s 4-D maths? How can you do such maths—and how can it represent something as concrete as an object rotating in ordinary space? It’s quite a long tale, because it took a very long time for mathematicians to learn to think abstractly—to free their imaginations from the tangible. And to do this, first they had to learn to think symbolically—to move from concrete arithmetic and geometry into the world of abstract, algebraic symbolism. So, if you ever struggled with algebra, take heart: this long history shows that generations of famous mathematicians have done so, too.

代数花了很长时间才从算术和几何中分离出来，成为一门独立的学科。直到中世纪，它才有了名字。时代，这要归功于九世纪波斯数学家穆罕默德·伊本穆萨 (al-)Khwārizmī。（阿拉伯语前缀 al- 不是 Khwārizmī 波斯名字的一部分，但由于他在巴格达工作，因此他以 al-Khwārizmī 这个名字为人所知。）他在哈里发马蒙的开创性巴格达大学或“智慧之家”学习，当时伟大的阿拉伯语翻译运动正处于鼎盛时期：希腊语、印度语和其他古代手稿从蓬勃发展的伊斯兰帝国的各个角落收集起来并翻译成阿拉伯语。帝国主义很少是道德的，而且往往是暴力的，但它最终会导致文化的相互影响，在这种情况下，富有远见的翻译运动非常重要，以至于到了 12 世纪，欧洲人开始学习阿拉伯语，以便将这些手稿翻译成拉丁语——包括托勒密的《天文学大成》和欧几里得的《几何原本》 ，以及新的阿拉伯语作品，如花拉子米的作品。“代数”这个名字来自他的书《Al-Jabr wa'l muqābalah》标题的第一个词——意思是类似《完成和平衡计算简明书》⁶

It took a long time for algebra to emerge from arithmetic and geometry as a separate subject. It didn’t even get its name until medieval times, and that was thanks to the ninth-century Persian mathematician Mohammed ibn-Mūsā (al-)Khwārizmī. (The Arabic prefix al- was not part of Khwārizmī’s Persian name, but since he worked in Baghdad, al-Khwārizmī is the name by which he became known.) He studied at Caliph al-Ma’mūn’s pioneering Baghdad-based university, or “House of Wisdom,” when the great Arabic translation movement was at its height: Greek, Indian, and other ancient manuscripts were being collected from all corners of the burgeoning Islamic empire and translated into Arabic. Imperialism is rarely ethical and often violent, but it can ultimately lead to cultural cross-fertilisation, and in this case the visionary translation movement was so important that by the twelfth century, Europeans were learning Arabic in order to translate these manuscripts into Latin—including Ptolemy’s Almagest and Euclid’s Elements, along with new Arabic works such as those of al-Khwārizmī. The name “algebra” famously comes from the first word in the title of his book Al-Jabr wa’l muqābalah—which means something like “The Compendious Book on Calculation by Completion and Balancing.”⁶

从花拉子米所附的问题来看，他所说的“完成”的一个例子是“完成平方”，这是你在学校里可能学到的解决二次方程的方法。这类方程在汉密尔顿虚数的故事中发挥了作用。例如，考虑 x ² + 1 = 0。今天的学生可以立即写下解：x = ± i，其中i代表“虚”数$\sqrt{-1}$⁷法国人勒内·笛卡尔在十七世纪将“虚数”一词引入数学。他指出，“对于我们能想象的任何方程”，都有可能找到解，但“在许多情况下，并不存在与我们想象的相对应的量。”换句话说，他是说你可以“想象”负数平方根的解，但它们并不存在——至少不是传统意义上的数字。⁸

Judging from the problems al-Khwārizmī included, an example of what he meant by “Completion” is “completing the square,” the method you might have learned in school to solve quadratic equations. Such equations play a part in the story of Hamilton’s imaginary numbers. For example, consider x² + 1 = 0. Today’s students can write down the solution immediately: x = ±i, where i stands for the “imaginary” number $\sqrt{-1}$.⁷ Frenchman René Descartes introduced the term “imaginary” to mathematics in the seventeenth century. He noted that “for any equation we can imagine” it is possible to find solutions, and yet, “in many cases no quantity exists that corresponds to what one imagines.” In other words, he was saying that you can “imagine” solutions with square roots of negative numbers, but they don’t really exist—at least, not as numbers in the traditional sense.⁸

如果笛卡尔认为这样的数字不存在，那么八百年前，在花拉子密的时代，-1 的平方根被认为是“不可能的”，而得出它的方程式被认为是无解的，这也就不足为奇了。因此，和大多数古人一样，花拉子密只关注有正解的二次方程，因为即使是负数似乎也没有实际意义。（七世纪的印度数学家婆罗摩笈多远远领先于他的时代，因为他考虑了正解和负解。）

If Descartes thought such numbers didn’t exist, then it’s not surprising that eight hundred years earlier, in al-Khwārizmī’s day, the square root of −1 was considered “impossible,” and the equations that gave rise to it were deemed unsolvable. So, like most of the ancients, al-Khwārizmī focussed only on quadratic equations with positive solutions, for even negative numbers seemed to have no practical meaning. (The seventhcentury Indian mathematician Brahmagupta was way ahead of his time, for he considered both positive and negative solutions.)

花拉子米也没有用我们今天使用的符号形式写方程式。事实上，在现代人看来，他的书更多的是算术而不是代数，当它被翻译成拉丁文时，它在欧洲产生的一个重要影响是普及了印度-阿拉伯十进制计数系统，该系统最终演变成我们现代的计数系统。然而，花拉子米经常被称为“代数之父”。他可能使用了文字而不是符号，他所包含的问题可能很简单——他告诉我们，他的目的是教学生如何解决“继承、遗产、分割、诉讼和贸易案件以及他们彼此之间的所有交易，或土地测量、运河挖掘、几何计算和其他对象”中的基本问题涉及各种类型和种类。”但他系统地列出了文字形式的线性和二次方程，以及解决这些方程的算法方法——即找到“未知数”，即我们现代的x和y。事实上，英文单词“算法”——意思是一组用于执行计算或其他操作的规则——来自“algorismi”，这是 al-Khwārizmī 的早期拉丁化尝试^。9

Al-Khwārizmī didn’t write equations in the symbolic form we use today, either. In fact, to modern eyes his book is more arithmetical than algebraic, and one of its important impacts in Europe, when it was translated into Latin, was the popularisation of the Hindu-Arabic decimal system of numeration that eventually evolved into our modern one. Yet Al-Khwārizmī is often called the “father of algebra.” He may have used words rather than symbols, and the problems he included may have been simple—his purpose, he tells us, was to teach students how to solve basic problems in “cases of inheritance, legacies, partitions, lawsuits and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts and kinds are concerned.” But he systematically set out word-form linear and quadratic equations, with algorithmic methods for solving them—that is, for finding the “unknown numbers,” our modern x’s and y’s. In fact, the English word “algorithm”—meaning a set of rules for performing a calculation or other operation—comes from “algorismi,” an early Latinised attempt at al-Khwārizmī.⁹

“代数之父”这样的称号让我不禁想问，是否也有一位“母亲”？花拉子米并非凭空而来——甚至连牛顿、麦克斯韦或爱因斯坦都不是凭空而来。花拉子米的大多数前辈都已湮没于历史长河之中，但或许其中也曾有女性参与其中。事实上，神秘的希帕蒂娅可能是我们所知的最接近“母亲”的人，尽管我们无法得知她有多大的独创性，因为她的作品只留下了碎片。但当时的信件表明，她确实写过一篇学术评论，评论了另一位“代数之父”称号的竞争者，即三世纪讲希腊语的亚历山大·丢番图，丢番图在解决代数应用题时使用单词缩写，在符号的发展过程中迈出了重要的一步。

Such an epithet as “father of algebra” makes me want to ask, was there a “mother,” too? Al-Khwārizmī didn’t arise in a vacuum—not even Newton or Maxwell or Einstein did. Most of al-Khwārizmī’s predecessors have been lost to history, but perhaps there were women involved somewhere along the line. As it is, the mysterious Hypatia may be the closest we have to a known “mother,” although the extent of her originality is not known, for only fragments of her work remain. But letters from the time show that she did write a learned commentary on another contender for the title of “father of algebra,” the third-century Greek-speaking Alexandrian Diophantus, who took a significant step in the process of developing symbols when he used word abbreviations in setting out algebraic word problems.

当然，在现代，我们也有一些优秀的女性代数学家，包括西澳大利亚大学的名誉教授谢丽尔·普雷格，她是许多年轻女性的当代数学榜样。再往前追溯一点，有开创性的埃米·诺特，爱因斯坦的同事。她被称为“现代代数之母”，因为她在现代代数概念方面的工作远远超出了我在这里谈论的学校水平的代数。半个世纪前——再举一位数学先驱女性——1872 年，玛丽·萨默维尔在去世前一天还在研究汉密尔顿的四元数，当时她将近 92 岁。她曾是数学天文学最新发展的著名专家，被她的同时代人称为“科学女王”。近年来，她再次受到人们的赞誉，包括——通过民众投票，击败同为苏格兰人的马克斯韦尔——成为 2017 年苏格兰皇家银行发行的十英镑聚合物钞票的封面人物。

In modern times, of course, we’ve had some fine female algebraists, including the University of Western Australia’s Emeritus Professor Cheryl Praeger, who has been a contemporary mathematical role model for many young women. Moving back in time a little, there’s the groundbreaking Emmy Noether, a colleague of Einstein. She has been dubbed the “mother of modern algebra,” for her work on modern algebraic concepts that go way beyond the school-level algebra I’m talking about here. Half a century earlier—to take just one more pioneering mathematical woman—in 1872 Mary Somerville had been studying Hamilton’s quaternions the day before she died, at the age of nearly ninety-two. She had been a famous expert on the latest developments in mathematical astronomy, and was known to her contemporaries as the “Queen of Science.” She has become celebrated again in recent times, including—by popular vote, and beating fellow Scot Maxwell for the honour—as the face of the polymer ten-pound note issued by Royal Bank of Scotland in 2017.

说到更遥远的过去，我最喜欢的“代数之父”是神秘的伊丽莎白时代数学家托马斯·哈里奥特。我并不孤单：英国代数学家詹姆斯·约瑟夫·西尔维斯特在 1883 年写给阿瑟·凯莱（汉密尔顿四元数的早期崇拜者）的一封信中，将哈里奥特描述为“现代代数之父”。哈里奥特也受益于早期数学家，但他死后出版的《分析艺术实践》是第一本完全以符号形式编写方程式的代数教科书，基本上使用了现代符号。它出版于 1631 年，比花拉子密晚了大约八个世纪，比丢番图晚了十四个世纪。这表明数学家花了多长时间才学会符号化思考。

As far as the more distant past goes, my favourite “father of algebra” is the enigmatic Elizabethan mathematician Thomas Harriot. I’m in good company: in an 1883 letter to Arthur Cayley—an early admirer of Hamilton’s quaternions—the British algebraist James Joseph Sylvester described Harriot as “the father of current algebra.” Harriot, too, was indebted to earlier mathematicians, but his posthumous Artis Analyticae Praxis (Practice of the Analytical Art) is the first algebra textbook where the equations are written entirely symbolically, using essentially modern symbols. It was published in 1631, some eight centuries after al-Khwārizmī and fourteen centuries after Diophantus. That’s an indication of how long it took for mathematicians to learn to think symbolically.

哈里奥特本人没有发表任何数学和实验成果：他的第一位赞助人是风趣而又备受争议的沃尔特·雷利爵士，而他则忙于航行公海、躲避异教徒猎人，最重要的是，开发新的数学应用来协助雷利的航海事业。（在伊丽莎白时代，拼写尚未固定，以沃尔特爵士命名的美国小镇被拼写为 Raleigh。他本人也使用这种拼写，但他最常用的是 Ralegh，所以这是今天许多学者更喜欢的拼写。）

Harriot himself didn’t publish any of his mathematical and experimental work: with the colourful and controversial Sir Walter Ralegh as his first patron, he was too busy sailing the high seas, dodging heretic hunters, and, above all, developing new mathematical applications to aid Ralegh’s navigational enterprises. (In Elizabethan times, spelling was not yet fixed, and the American town named after Sir Walter is spelled Raleigh. The man himself used this spelling, too, but most often he used Ralegh, so this is the spelling many scholars prefer today.)

尽管哈里奥特从未发表过他的发现，但他留下了数千页手写的计算和观察记录——他在收集的大量数据和方程式中寻找隐藏的模式和普遍性时不断思考。这确实是代数的一个关键点：模式和普遍性。符号方程的美妙之处在于，当你一眼就能看到问题时，就会更容易看到这些普遍模式。比较一下：

Although Harriot never got around to publishing his discoveries, he left behind thousands of pages of handwritten calculations and observations— his thoughts in progress as he searched for hidden patterns and generalities in the vast amounts of data and equations he’d collected. And that is, indeed, a key point of algebra: pattern and generality. The beauty of symbolic equations is that it’s much easier to see these general patterns when you can see a problem at a glance. Compare this:

取未知数的平方，然后将未知数与其自身相加，并从平方中减去和；现在总数为八。

Take the square of the unknown number, then add the unknown number to itself and take the sum away from the square; now let the total be eight.

替换为：

with this:

x ² – 2 x = 8。

x² – 2x = 8.

不仅如此，最早的数学家们分别求解每个方程，但如果你能看到适用于方程x ² – 2 x = 8 的任何方法也适用于任何相同形式的方程x ² – ax = b ，那么就更容易了。最终，古代数学家们开始认识到这一点，但进展相对较慢，因为他们必须把所有这些模式记在脑子里，或者记在长而复杂的句子里，很容易忘记。

And there’s more: the earliest mathematicians solved each equation separately, but it’s easier if you can see that whatever method works for the equation x² – 2x = 8 will also work for any equation of the same form, x² – ax = b. Eventually, ancient mathematicians did begin to recognise this, but progress was relatively slow because they had to keep all these patterns in their heads, or in long, convoluted sentences, and it was easy to lose track.

第一个以透明、可识别的现代符号形式发表方程式的人是哈里奥特 (Harriot) 的遗嘱执行人，他们于 1631 年发表了方程式，然后是笛卡尔 (Descartes)，他于 1637 年在《方法论》的附录中发表^了方程式。10 （之前也有过几次尝试，但符号系统——更恰当地说是缩写——很扭曲，也很特殊。）¹¹甚至我们认为理所当然的 +、−、= 和 × 符号也只是在 17 世纪才开始广泛使用。这意味着，我们所知道的早期代数学家——古代美索不达米亚人、埃及人、中国人和希腊人、中世纪的印度人、波斯人和阿拉伯人，以及近代欧洲人——都主要用文字或图形来表达他们的方程式。

The first to publish any equation in a transparent, recognisably modern symbolic form were Harriot’s executors in 1631, and then Descartes in an appendix to his 1637 Discourse on Method.¹⁰ (There were a few earlier attempts, but the symbolism—more properly called abbreviation—was tortured and idiosyncratic.)¹¹ Even the +, −, =, and × signs we take for granted only came into widespread use in the seventeenth century. Which means that the earlier algebraists we know of—the ancient Mesopotamians, Egyptians, Chinese, and Greeks, the medieval Indians, Persians, and Arabs, as well as the early modern Europeans—all had expressed their equations mostly in words or pictorial word images.

• • •

• • •

符号思维是一种独特的技能，这一点从漫长的历史中可以看出来。以我上面给出的单词问题为例：它是算法思维的一个例子。但符号思维不仅仅是算法思维，因为它的符号有时包含着一种新创造力的种子——一种新的深远而又经济的思维。

It is a singular skill to think symbolically, as this long history shows. Take the word problem I gave above: it is an example of algorithmic thinking. But symbolic thinking is algorithmic and more, for its symbols sometimes contain the seeds of a new kind of creativity—a new kind of far-reaching yet economical thought.

一个经典的例子是爱因斯坦的E = mc ²。爱因斯坦并没有着手寻找能量和物质之间的联系。相反，他只是想根据他的新相对论计算运动电子的动能，以便可以通过实验检验他的理论预测。然而几个月后，26 岁的爱因斯坦开始意识到这个方程的重要性。他在 1905 年的第五篇开创性论文中写了这个方程，这是他的奇迹之年，但他又花了两年时间才梳理出这个符号关系的完整而戏剧性的含义。意识到这不仅仅是对特定形式的能量和特定类型的物质的计算，它是普遍的：如果一个物体获得（或失去）能量，也会获得（或失去）质量。这个奇怪的想法与我们所有的常识经验格格不入——但它就在那里，隐藏在他的方程符号中。实验物理学家花了几十年的时间才通过实验证实了这个惊人的数学预测^。12

A classic case is Albert Einstein’s E = mc². Einstein did not set out to find the connection between energy and matter. Rather, he simply wanted to calculate the kinetic energy of a moving electron according to his new theory of relativity, so that his theoretical prediction could be tested experimentally. A few months later, however, twenty-six-year-old Einstein began to realise the significance of his equation. He wrote it up in his fifth groundbreaking paper of 1905, his annus mirabilis, but it would take him two more years to tease out the full, dramatic implications of this symbolic relationship. To realise that this wasn’t just a calculation about a particular form of energy and a particular type of matter, it was general: if a body gains (or loses) energy, it also gains (or loses) mass. This bizarre idea is alien to all our commonsense experience—but there it was, hidden in the symbols of his equation. It took experimental physicists decades to experimentally confirm this astonishing mathematical prediction.¹²

一个更简单、更早的例子是幂序列x、x ²、x ³等。第一个“幂”是 1，所以x实际上是x ¹，其中 1 传统上在几何上与一维线相关。接下来的两个幂x ²和x ³发音为“ x平方”和“ x立方”，类比正方形的面积和立方体的体积。这些名称突显了早期数学家以几何方式而非代数方式思考的方式，因为几何具有有形的性质。相比之下，符号代数是抽象的：你必须赋予它意义，即使它只是显示一个有趣的模式，例如x、x ²、x ³、x ⁴ ……但这种灵活性是代数的强大之处。你可以写下任意数量的（有限）高次幂，而不必将它们可视化为物理对象。

A much simpler and earlier example is the sequence of powers x, x², x³, and so on. The first “power” is 1, so x is really x¹, where the 1 was traditionally linked geometrically to a 1-D line. The next two, x² and x³, are pronounced “x squared” and “x cubed” by analogy with the area of a square and the volume of a cube. These names highlight the way that early mathematicians thought geometrically rather than algebraically, because of the tangible nature of geometry. By contrast, symbolic algebra is abstract: you have to give it meaning, even if it is simply the display of an interesting pattern such as x, x², x³, x⁴, … But this flexibility is algebra’s great strength. You can write down as many (finite) higher powers as you like, without having to visualise them as physical objects.

今天看来，这似乎显而易见，但数学家们花了三千五百年的时间才从解决二次方程（“二次”源于拉丁语“平方”，因此二次方程的最高幂是x ²（古人称之为未知数乘以自身）——到解决“三次”和更高次方程。当然，这些高次方程要困难得多；但解法不易的部分原因是代数长期以来与文字和具体图像紧密相关。

This may sound obvious today, but it took three and a half thousand years for mathematicians to move from solving quadratic equations— “quadratic” derives from the Latin for “square,” so quadratic equations are those whose highest power is x² (the unknown multiplied by itself, as the ancients put it)—to solving “cubic” and higher equations. These higherdegree equations are much more difficult, of course; but part of the reason solutions didn’t come easily was that algebra was tied to words and concrete images for such a very long time.

例如，我提到了花拉子米为了解决二次方程而采用的“配方法”。这实际上是一个有四千年历史的问题，可以追溯到（就历史记录而言）由生活在现代伊拉克地区的数学家制作的楔形文字板。这些古代美索不达米亚人通过配方法来解决二次方程。这是当时一个典型的教学问题：“将我的长度的 20 倍加到我的正方形面积上，[得到] 21。我的正方形有多方？” ¹³这类问题及其解决方法与今天教授的算法相似——只是四千年前，这种方法完全是几何学方法。首先，画一个正方形任意边长为x（现代符号）；然后向其中添加一个尺寸为 20 × x的矩形。现在将这个额外的矩形分成两个相等的小矩形，并将它们排列在原始正方形的旁边和下面。最后，完成这个新的更大的正方形，如图1.2所示。

For instance, I mentioned al-Khwārizmī’s “completing the square” in order to solve a quadratic equation. It’s actually a four-thousand-year-old problem, dating back (as far as the historical record shows) to cuneiform tablets made by mathematicians living, like al-Khwārizmī, in the region of modern-day Iraq. These ancient Mesopotamians solved quadratic equations by literally completing a square. Here is a typical teaching problem of the time: “Add 20 of my length to the area of my square, [to get] 21. How square is my square?”¹³ This type of problem, and the algorithm for solving it, is similar to those taught today—except that four millennia ago, the method was worked out entirely geometrically. First, draw a square of arbitrary side x (in modern notation); then add to it a rectangle of dimensions 20 × x. Now split this additional rectangle into two equal smaller ones and arrange them beside and below the original square. Finally, complete this new, larger square, as in figure 1.2.

美索不达米亚人在开发这种方法时，至少在最初阶段，考虑到了实际问题。他们生活在水资源极为稀缺的地区，他们的石碑上记载了许多与运河和水库开挖、蓄水池容量、水坝和堤坝的修建以及与这些任务相关的行政账目有关的问题——为了解决这些问题，这些古代数学家必须解出与面积和体积有关的方程式。14^近三千年后，花拉子密也关注类似的实际问题，他使用了类似的几何方法来完成平方——直到十七世纪，其他数学家也是如此。

The Mesopotamians had practical problems in mind when they developed this method, at least initially. Living in a land where water was at a premium, their tablets contain many problems relating to canal and reservoir excavations, the capacity of cisterns, the construction and repair of dams and levees, and administrative accounts relating to these tasks—and to solve these problems, these ancient mathematicians had to solve equations relating to areas and volumes.¹⁴ Nearly three thousand years later, al-Khwārizmī, too, focussed on similar practical problems, and he used a similar geometrical method of completing the square—and so did other mathematicians right up to the seventeenth century.

伊斯兰数学家沙拉夫丁·塔西是最早在寻找三次方程解方面取得进展的人之一，大约在公元 1200 年，但第一个发表正确的通用三次算法的人是意大利数学家吉罗拉莫·卡尔达诺，发表在他 1545 年出版的《大术》一书中。15^和他之前的所有人一样，他仍然用文字（或文字缩写）而不是符号来写出他的解，并且他仍然以几何方式设计他的方法——通过令人惊叹的可视化壮举真正完成一个立方体。

The Islamic mathematician Sharaf al-Dīn al-Ṭūsī was one of the earliest to make progress in the search for solutions of cubic equations, in about 1200 CE, but the first to publish correct general cubic algorithms was the Italian mathematician Girolamo Cardano, in his 1545 book Ars Magna (The Great Art).¹⁵ Like everyone before him, he still wrote his solutions in words (or abbreviations of words) rather than symbols, and he still devised his method geometrically—literally completing a cube in a stunning feat of visualisation.

几年后，当塔塔利亚仍保守秘密时，卡尔达诺发现西皮奥内·德尔·费罗也比塔塔利亚早找到了解决方案。因此，卡尔达诺觉得他可以违背诺言，发表论文——他总是在寻找宣传机会。但他完全承认这两个人，并超越他们，解决了更广泛、更普遍的方程式。尽管如此，塔塔利亚还是很生气——以至于他向卡尔达诺发起公开决斗，不是用剑，而是用解题比赛。卡尔达诺谨慎地拒绝了：在这种竞争激烈的文艺复兴盛会中，名誉（和工作）很容易赢得和失去。此外，塔塔利亚已经与德尔·费罗的学生安东尼奥·菲奥尔较量过，后者知道他老师的三次方法——而塔塔利亚赢得了那场比赛。

Some years later, while Tartaglia was still holding onto his secret, Cardano discovered that Scipione del Ferro had also found the solution, before Tartaglia. So, Cardano felt he could break his promise and publish—he always had his eye on a publicity opportunity. But he fully acknowledged both men, and he went beyond them in solving a wider, more general range of equations. Still, Tartaglia was furious—so much so that he challenged Cardano to a public duel, not with swords but with a problem-solving competition. Cardano prudently refused: reputations (and jobs) were easily won and lost in these fiercely competitive Renaissance spectacles. Besides, Tartaglia had already taken on del Ferro’s student Antonio Fior, who knew of his teacher’s cubic method—and Tartaglia had won that match.

在书中，卡尔达诺用一页巧妙的几何类比解释了他的通用算法，然后给出了具体的说明性例子。他是这样解释他求解x ³ = 6 x + 40 的方法的（使用现代符号，我也将使用现代符号使卡尔达诺的算法更容易理解；请耐心听我说，即使你只是浏览一下，因为最后一行的表达式形式与虚数和向量的故事有着惊人的相关性）：“将 2（x的系数的三分之一）的立方乘以 8；用 400（20 的平方）减去这个数字，[这是常数的一半，等于 392；它的平方根加上 20 等于$20+\sqrt{392}$并从 20 个品牌中减去$20-\sqrt{392}$；这些数的立方根之和，$\sqrt[3]{20+\sqrt{392}}+\sqrt[3]{20-\sqrt{392}}$，是x的值。”呼！你不得不佩服他的耐心，想出了这么复杂的问题^。16

In his book, Cardano explained his general algorithm in a page of ingenious geometrical analogy and then gave specific illustrative examples. This is how he explained his method for solving x³ = 6x + 40 (to use modern notation, which I’ll also use to make Cardano’s algorithm a little easier to follow; bear with me, even if you just skim through it, because the form of the expression in the last line has surprising relevance to the story of imaginary numbers and vectors): “Raise 2, one-third the coefficient of x, to the cube, which makes 8; subtract this from 400, the square of 20, [which is] one-half of the constant, making 392; the square root of this added to 20 makes $20+\sqrt{392}$, and subtracted from 20 makes $20-\sqrt{392}$; and the sum of the cube roots of these, $\sqrt[3]{20+\sqrt{392}}+\sqrt[3]{20-\sqrt{392}}$, is the value of x.” Phew! You’ve got to admire his patience in coming up with something so convoluted.¹⁶

从向量发展史和数学发展史的角度来看，有趣的是，当这种解的平方根号下的数字为负时会发生什么。也就是说，当你有一个虚数，例如$\sqrt{-121}$。

The interesting thing, from the point of view of the story of vectors— and of the development of mathematics in general—is what happens when the number under the square root sign in such a solution is negative. That is, when you have an imaginary number such as $\sqrt{-121}$.

美索不达米亚人忽略了二次方程的负解和虚解，因为它们与他们试图解决的实际问题无关——田地和运河的维度不能为负数或虚数。希腊人也是如此，直到花拉子米和塔西，直到卡尔达诺被迫与这些“不可能”的数字搏斗。他研究方程数学只是为了它本身，为了智力挑战——他被这样一个事实所困扰：如果他用他用于x ³ = 6 x + 40 的相同方法将其应用于x ³ = 15 x + 4，那么x的值将是

The Mesopotamians had ignored negative and imaginary solutions of quadratic equations because they had no relevance to the practical problems they were trying to solve—you can’t have negative or imaginary dimensions of fields and canals. Similarly for the Greeks, through to al-Khwārizmī and al-Ṭūsī, and right up until Cardano was forced to wrestle with these “impossible” numbers. He was studying the mathematics of equations simply for its own sake, for the intellectual challenge of it—and he was flummoxed by the fact that if he took the same method he’d used for x³ = 6x + 40 and applied it to x³ = 15x + 4, then the value of x turned out to be

$\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$。

$\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$.

卡尔达诺的结论是，这样的解决方案是“复杂的”，“既微妙又无用”——因为除了不受欢迎的$\sqrt{-121}$，他已经知道事实上x = 4。他知道这一点是因为他会通过猜测解决方案来开始理解问题——数学家们一直在做的事情。当没有已知的解决问题的算法时，这种方法尤其有用，所以它是古代代数的开始。对于卡尔达诺方程x ³ = 15 x + 4，你可以通过尝试一个简单的可能值（例如x = 3）来了解这个想法；比较每一边你会发现这个值太小了，所以试试x = 4。在这种情况下它马上就起作用了，但有时你必须尝试中间值。这仍然是数学家“数字化”解决难题的方式，尽管他们有算法（现在还有计算机）来有效和详尽地选择他们的猜测。

Cardano concluded that such a solution was “sophistic,” and “as subtle as it is useless”—because aside from the unwelcome $\sqrt{-121}$, he already knew that in fact x = 4. He knew this because he would have begun to understand the problem by guessing the solution—something mathematicians have always done. It is especially useful when there isn’t a known algorithm for solving a problem, so it is the way ancient algebra began. For Cardano’s equation x³ = 15x + 4, you can see the idea by trying a simple possible value such as x = 3; comparing each side you see that this is too small, so try x = 4. In this case it works straightaway, but sometimes you have to try intermediate values. This is still the way mathematicians solve difficult problems “numerically,” although they have algorithms (and now computers) to choose their guesses efficiently and exhaustively.

十五年后，大约 1560 年，又出现了另一个优秀的早期现代意大利代数学家拉斐尔·邦贝利（Rafael Bombelli）重新审视了卡尔达诺的难题。问题是，$十=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$考虑到解是x = 4，这和它有什么关系？经过一番思考后，Bombelli 突然有了所谓的“疯狂想法”：如果你能将$\sqrt{-121}$像这样：$\sqrt{121}\times \sqrt{-1}$， 要得到$11\sqrt{-1？}$那么，你能找到现在所谓的“复数”吗？复数是实数和虚数的混合，它的立方是$2+11\sqrt{-1}？$令人惊讶的是，经过反复尝试和耐心等待，他发现$2+\sqrt{-1}$是$\sqrt[3]{2+11\sqrt{-1}}$，你可以看到如果你乘以${\left(2+\sqrt{-1}\right)}^3$。同样，他发现$2-\sqrt{-1}$是$\sqrt[3]{2-11\sqrt{-1}}$按照 Cardano 的解决方案将它们加在一起，你会得到

Fifteen years later, around 1560, yet another excellent early modern Italian algebraist, Rafael Bombelli, took another look at Cardano’s conundrum. The question was, what did $x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$ have to do with it, given that the solution was x = 4? After a great deal of thinking Bombelli suddenly had what he called “a wild thought”: what if you could factor $\sqrt{-121}$ like this: $\sqrt{121}\times \sqrt{-1}$, to get $11\sqrt{-1?}$ And could you then find what is nowadays called a “complex” number—a mix of real and imaginary numbers—whose cube is $2+11\sqrt{-1}?$ Amazingly, with trial, error, and a lot of patience he found that $2+\sqrt{-1}$ is a solution of $\sqrt[3]{2+11\sqrt{-1}}$, as you can see if you multiply out ${\left(2+\sqrt{-1}\right)}^3$. Similarly, he found that $2-\sqrt{-1}$ is a solution of $\sqrt[3]{2-11\sqrt{-1}}$. Adding these together as in Cardano’s solution, you get

$十=2+\sqrt{-1}+2-\sqrt{-1}$，

$x=2+\sqrt{-1}+2-\sqrt{-1}$,

看似奇迹般地，得出了x = 4。谜团解开了！

which, seemingly miraculously, gives x = 4. Mystery solved!

不过，它只针对这个特殊情况得到了解决，当邦贝利事先知道x = 4 时——他对操纵虚数有着绝妙的洞察力，但他没有通用算法。他也没有用我在这里给出的透明现代符号形式来写他的方程式——而且像卡尔达诺一样，他贬低$\sqrt{-1}$他认为这个奇怪的数字“诡辩”。但当他的著作《代数学》于 1570 年代出版时，他确实将这个奇怪的数字更坚定地放在了数学雷达上。当时，他或其他任何人都不知道它会变得多么有用——在汉密尔顿的四元数和向量中，以及在工程、计算和量子理论等领域^。17

It was solved only for this special case, though, and when Bombelli knew beforehand that x = 4—he’d had a brilliant insight about manipulating imaginary numbers, but he had no general algorithm. He didn’t write his equations in the transparent modern symbolic form I’ve given here, either—and like Cardano, he disparaged $\sqrt{-1}$ as “sophistic.” But he did put this strange number more firmly on the mathematical radar when his book Algebra was published in the 1570s. Little did he or anyone else know back then just how useful it would become—in Hamilton’s quaternions and vectors, and in fields such as engineering, computing, and quantum theory.¹⁷

至于三次方程，哈里奥特于 1600 年左右首次发现了一般的符号代数解——而且他的证明中没有参考几何。约翰·沃利斯——也许是哈里奥特和牛顿时代之间最优秀的英国数学家——是少数几个承认哈里奥特将代数从几何中解放出来的成就的近现代人之一，用沃利斯的话来说，他“将代数视为纯粹的代数，从其自身的原理出发，不依赖于几何或与几何有任何联系。” ¹⁸这为我将要讨论的汉密尔顿奠定了一些数学背景，因为他也想找到一种纯代数表示几何运算的方法——在他看来，不是立方体而是空间旋转，这个问题的解将使他发明向量。

As for cubic equations, it was Harriot who first found general, symbolic algebraic solutions, sometime around 1600—and with no reference to geometry for his proofs. John Wallis—perhaps the best British mathematician between Harriot’s time and Newton’s—was one of the few nearcontemporaries to recognise Harriot’s achievement in liberating algebra from geometry, treating, as Wallis put it, “algebra purely by itself, and from its own principles, without dependence on geometry, or any connexion therewith.”¹⁸ This sets some mathematical context for where I’ll be going with Hamilton, for he, too, wanted to find a way to represent purely algebraically a geometrical operation—in his case, not cubes but rotations in space, the problem whose solution will lead him to invent vectors.

用代数来设想几何不仅扩展了代数，也扩展了几何，我们将看到，随着向量和张量的出现，这两种数学齐头并进，相互影响。但第一步是像哈里奥特和沃利斯那样，认识到代数本身就是一门学科，就像几何一样。

Using algebra to envisage geometry expands not just algebra but geometry, too, and we’ll see that these two kinds of maths went hand in hand, each influencing the other, as vectors and tensors emerged. But the first step had been to see, as Harriot and Wallis did, that algebra was a subject in its own right, just as geometry was.

哈里奥特的灵感来自多才多艺的早期现代法国人弗朗索瓦·韦达，后者开始使用大写字母表示未知数，哈里奥特刻苦钻研其关于三次方程的论文。哈里奥特像我们今天一样使用小写字母，他如此全面地使用符号，以至于他成为了符号思维大师。他的一个洞见是表明多项式方程可以通过乘以它们的因子来生成——例如，两个线性因子生成一个二次方程，三个线性因子生成一个三次方程，四个线性因子生成一个四次方程，等等。这个“因式定理”现在看起来可能很明显——你可能在高中代数课上学过它——但在哈里奥特之前没有人写过这样的符号方程

Harriot had taken his lead from the versatile early modern Frenchman François Viète, who had begun to use uppercase letters for unknowns, and whose treatise on cubic equations Harriot studied assiduously. Harriot used lowercase letters as we do today, and he used symbols so completely that he became a master of symbolic thinking. One of his insights was to show that polynomial equations can be generated by multiplying their factors—for example, two linear factors generate a quadratic, three give a cubic, four a quartic, and so on. This “factor theorem” may seem obvious now—you may have learned it in a senior high school algebra class—but no one before Harriot had written symbolic equations such as

$（十-升）（十-米）（十-n）=0.$

$(x-l)(x-m)(x-n)=0.$

实际上，哈里奥特并没有使用单独的圆括号来表示乘积，而是将因数一个接一个地写在上方，并用方括号括住群。他倾向于使用a而不是x来表示未知数，使用aa而不是a ² 。x 、x ²、x ³、x ⁴ 、...这些符号是笛卡尔于 1637 年发表的，尽管他有时仍像哈里奥特一样使用xx甚至aa。无论如何，这个方程暗示的是，三次方程必须有三个解，x = l、x = m、x = n，无论它们是正数还是负数，实数还是虚数。相比之下，卡尔达诺的算法提到“唯一”解时，好像只有一个——如果你用物质立方体来想象它，你就会期望它只有一个^。19

Actually, Harriot didn’t use separate round brackets for products, but wrote the factors one on top of the other with a square bracket around the group. And he tended to use a rather than x for the unknown, and aa instead of a². We owe the notation x, x², x³, x⁴, … to Descartes, who published it in 1637, although he still sometimes used xx and even aa, like Harriot. Either way, what this equation hints at is that a cubic equation has to have three solutions, x = l, x = m, x = n, whether they are positive or negative, real or imaginary. By contrast, Cardano’s algorithm had spoken of “the” solution, as if there were only one—which is what you’d expect if you were imagining it in terms of a material cube.¹⁹

要了解哈里奥特符号体系的优势（它与我在这里使用的现代版本没有太大区别，并且同样透明），请考虑从邦贝利的x = 4 的出发点来求解那个棘手的卡尔达诺方程。哈里奥特的方法建议首先将x ³ = 15 x + 4 写为x ³ − 15 x − 4 = 0。这正是你在高中时会做的事情，然后将x ³ − 15 x − 4 除以x − 4 得到x ³ − 15 x − 4 =（x − 4）（x ² + 4 x + 1）。当x = 4或x ² + 4 x + 1 = 0时，该式等于零。你可以完成平方来求解二次方程，以找到另外两个解，$十=-2+\sqrt{3}$和$十=-2-\sqrt{3}$，总共有三个解。在这种情况下，所有解都是实数，而 Cardano 的复杂表达式$十=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$似乎没有。……然而，后来的数学家们通过发现复数本身实际上有三个立方根，将历史点联系起来。因此，卡尔达诺的棘手方程的三个实数解可以从他的算法中恢复出来！我在尾注中展示了如何做到这一点^。20

To see the advantage of Harriot’s symbolism, which was not too different from the modern version I’m using here, and just as transparent, consider solving that pesky Cardano equation from Bombelli’s starting point of knowing that x = 4. Harriot’s method suggests that first you write x³ = 15x + 4 as x³ − 15x − 4 = 0. This is just what you would have done in high school, and you’d then divide x³ − 15x − 4 by x − 4 to get x³ − 15x − 4 = (x − 4)(x² + 4x + 1). This equals zero when x = 4 or when x² + 4x + 1 = 0. You can complete the square to solve the quadratic, to find two additional solutions, $x=-2+\sqrt{3}$ and $x=-2-\sqrt{3}$, making a total of three solutions. In this case, all the solutions turn out to be real, and Cardano’s complicated expression $x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$ doesn’t come into it. Or so it seems.… Later mathematicians, however, would connect the historical dots by discovering that in fact complex numbers themselves each have three cube roots. So, the three real solutions of Cardano’s pesky equation can be recovered from his algorithm! I’ve shown how in the endnote.²⁰

他想表达的是，用符号进行概括比用文字容易得多。当你能够概括时——当你能够看到适用于出乎意料的广泛问题的共同模式时——你就可以在科学技术和数学上取得非凡的进步。例如，麦克斯韦能够展示光的电磁波性质，并预测无线电波的存在，因为他对电磁学的数学分析得出了与描述拨动吉他或小提琴弦时看到的波形相同的方程。诺特巧妙地概括了对称性的数学模式与能量和动量等物理量守恒之间的关系。

What he was getting at is that generalisation is far easier in symbols than in words. And when you can generalise—when you can see common patterns that apply to an unexpectedly wide range of problems—you can make extraordinary progress in science and technology as well as mathematics. For instance, Maxwell was able to show the electromagnetic wave nature of light, and to predict the existence of radio waves, because his mathematical analysis of electromagnetism turned up the very same kind of equation that had been used to describe the wave pattern you see when you pluck a guitar or violin string. And Noether brilliantly generalised the relationship between mathematical patterns of symmetry and the conservation of physical quantities such as energy and momentum.

稍后将详细介绍这些例子；与此同时，哈里奥特学者穆里尔塞尔特曼巧妙地总结了哈里奥特的重要性和代数符号的力量：

More on these examples later; meantime, Harriot scholar Muriel Seltman sums up neatly both Harriot’s importance and the power of algebraic symbolism:

符号和数学思维过程之间存在着相互关系，很难高估哈里奥特的技巧和清晰的思维效果，这些技巧和清晰的思维通过符号来指导你的工作，从而使数学以一种全新的方式变得易于理解……可视化是显而易见的，但非常重要。现在可以将符号当作不可可视化的概念来操纵，符号只是其具体化^。22

There is a reciprocal relation between symbolism and mathematical thought-processes, and it would be hard to overestimate the effect of Harriot’s techniques and clarity of thought expressed in a symbolism that directs what you do visually and therefore makes mathematics accessible in a totally new way…. The visualizability is obvious but profoundly important. It is now possible to manipulate the symbol as if it were the non-visualizable concept of which the symbol is only the embodiment.²²

这也让我想到了最终要说的。汉密尔顿的四维四元数没有传统的具体几何类似物，我们大多数人也不可能想象爱因斯坦的四维时空（更不用说十维弦理论了）；但描述它们的数学符号方程却有自己的可触性。它们可以被操纵，就好像四维“真的”存在一样，因为从代数上讲，^x4与“正方形” x2和“立方体” ^x3一样有效^，而且——跳到坐标和向量——（a ，b，c，d ）与（ a，b，c ）是同一类东西。

Which brings me to where I’m ultimately heading. Hamilton’s fourdimensional quaternions don’t have traditional concrete geometrical analogues, and it’s impossible for most of us to visualise Einstein’s fourdimensional space-time (let alone such things as ten-dimensional string theory); but the mathematical symbolic equations that describe them have their own kind of tangibility. They can be manipulated as if four dimensions “really” do exist, because algebraically speaking x⁴ is just as valid as the “square” x² and the “cube” x³, and—jumping ahead to coordinates and vectors—(a, b, c, d) is the same kind of thing as (a, b, c).

随着代数符号的兴起，一种新的抽象思维应运而生，为微积分的发展开辟了道路，并最终为矢量和张量微积分的发展开辟了道路。这种奇妙的语言使数学家能够解决大量新的、更复杂的问题，这些问题的解决方案不仅为我们带来了新技术，也为我们提供了看待现实的新方式。因此，下一章将简要介绍微积分的非凡历史。

With the rise of algebraic symbolism, a new kind of abstract thinking arose, which opened the way for the development of calculus—and ultimately to vector and tensor calculus. This marvelous language has enabled mathematicians to solve a vast range of new, more complicated problems—problems whose solutions have given us new technologies as well as new ways of seeing reality. So the next chapter will offer a brief tour through the extraordinary story of calculus.

在布鲁姆大桥获得“电”学发现的十年前，汉密尔顿利用微积分做出了他的第一个重大发现：一种新的光学现象的理论存在——某些晶体中存在一种特殊类型的折射，以前从未有人观察到过。

A decade before his “electric” insight at Broome Bridge, Hamilton had used differential calculus to make his first major discovery: the theoretical existence of a new optical phenomenon—a special type of refraction in certain crystals, which no one had ever observed before.

光学是研究光的行为的学科，尽管没有人知道光到底是什么，但对于物理学家和数学家来说，光学都是一个令人兴奋的课题。然而，越来越多的物理学家相信，光不是牛顿所认为的粒子，而是某种波。麦克斯韦将把这一结论推向最终结论——强调向量微积分在这一过程中的重要性——但潮流在 1801 年开始转变，当时有一项非凡的实验，如今已成为物理学史上的经典：托马斯·杨的“双缝”实验。

Optics, the study of how light behaves, was an exciting topic for both physicists and mathematicians, even though no one knew what light actually was. Yet a growing number of physicists were confident that it wasn’t a particle, as Newton had believed, but a wave of some sort. Maxwell will take this to its ultimate conclusion—highlighting the importance of vector calculus in the process—but the tide had begun to turn in 1801, with a remarkable experiment that is now a classic in the story of physics: Thomas Young’s “double slit” experiment.

这个想法是为了观察两束光相互作用时会发生什么，并看看由此产生的相互作用模式是波还是粒子的模式。如果你把两块鹅卵石扔进池塘，你就能看到波产生的模式。石头撞击水面时产生的涟漪以每块石头为中心向外扩散，形成同心圆——你可以观察一块石头的涟漪跳过另一块石头周围的涟漪的地方。当它们跳过时，水位上升，所以两个波​​纹会结合在一起形成一个更大的波浪。如果你能用光做同样的实验——如果光确实以波的形式传播——你会看到这种效果，即两波光波相互加强，亮度增加。更难看到的是，在水面的某些地方，波纹会相互抵消——一个波的波峰与另一个波的波谷相遇——所以水面是平的。这种效果的光学类似物是一个暗斑——没有光，因为波相互抵消了。但光是波吗？这就是杨开始探索的。

The idea was to observe what happens when two beams of light interact with each other, and to see if the resulting interaction patterns were those of waves or particles. You can see the pattern that waves produce if you drop two pebbles into a pond. The impact of the stones as they hit the water produces ripples that spread out in concentric circles centred around each of the stones—and you can watch the places where the ripples from one stone jump over those around the other. And when they do, the water level rises, so the two ripples combine to produce a bigger wave. If you could do the same experiment with light—and if it did travel as a wave—you’d see this effect as an increase in brightness as the two light waves reinforced each other. It’s harder to see that at some places on the water surface the ripples cancel each other out—the peak of one wave has met up with the trough of the other—and so the water surface is flat. The optical analogue of this effect would be a dark patch—there’d be no light because the waves canceled each other. But was light a wave? That is what Young set out to explore.

他通过两个小针孔照射光线，产生两束光，然后让两束光照射到屏幕上，从而呈现出“干涉图样”。如果光是由粒子组成的，那么每束光都会笔直向前，因此你会看到两个明亮的斑块与两个针孔成一线——类似于两堆信件被扔进有两个开口的邮箱，或者两束乒乓球射向屏幕。然而，如果光以波的形式传播，那么它会在针孔周围弯曲（或“衍射”），产生像池塘上的涟漪一样的圆形波——而交叉的涟漪会在屏幕上产生明暗斑块的干涉图样。果然，这正是杨所发现的。这是一个美丽的实验。然而，一个世纪后，爱因斯坦提出了光量子的概念，后来被称为光子。光子不是牛顿式的物质粒子，但在与原子的相互作用中，它们可以表现出粒子般的行为。除了这一奇怪的发现之外，量子物理学家还改编了杨的实验，表明光子和电子等亚原子物质粒子都表现出波干涉图案。换句话说，光和物质都可以表现出波状或粒子状的行为，具体取决于情况。但这只是后见之明，所以让我们回到杨的光波动理论，以及汉密尔顿对它的贡献。

He created two beams of light by shining light through two tiny pinholes, and he manifested the resulting “interference pattern” by letting the two beams shine onto a screen. If light were made of particles, each beam would just go straight ahead, so you’d expect to see just two bright patches in line with the two pinholes—analogous to two piles of letters dropped into a postbox with two openings, or two streams of ping-pong balls fired at a screen. If light traveled as waves, however, then it would bend (or “diffract”) around the pinholes to produce circular waves like the ripples on the pond—and the crisscrossing ripples would produce an interference pattern of light and dark patches across the screen. And sure enough, this is just what Young found. It was a beautiful experiment. A century later, however, Einstein would introduce the idea of a light quantum, later called a photon. Photons are not Newtonian-style material particles, but in their interactions with atoms they can behave in a particle-like way. To add to this bizarre finding, quantum physicists have adapted Young’s experiment to show that both light photons and subatomic material particles such as electrons exhibit wave interference patterns. In other words, both light and matter can manifest wave-like or particle-like behavior, depending on the situation. But this is hindsight, so let’s return to Young’s wave theory of light, and what Hamilton did with it.

杨是玛丽·萨默维尔的朋友，她回忆说，起初英国很少有人认真对待杨的光波推论，因为没有人想贬低伟大的牛顿。（事实上，当时他有充分的理由拒绝波动理论。）但阻碍杨实验被接受的不仅仅是牛顿的名声：还有一个事实是，没有人知道光波可能是由什么构成的，或者它是如何从太阳到地球的旅程中穿越空旷的空间的。尽管如此，这并没有阻止 19 世纪早期的数学家，如汉密尔顿，从理论上探索光波——这是研究人员自牛顿时代的克里斯蒂安·惠更斯以来一直在做的事情^。1

Young was a friend of Mary Somerville, and she recalled that few in England took his light-wave deduction seriously at first, because no one wanted to diminish the great Newton. (In fact, at the time he’d had good reasons for rejecting a wave theory.) But it wasn’t just Newton’s reputation that got in the way of the reception of Young’s experiment: there was also the fact that no one knew what a light wave could possibly be made of, or how it managed to travel through empty space in its journey from the Sun to Earth. Still, this hadn’t stopped early nineteenth-century mathematicians such as Hamilton from exploring light waves theoretically—something researchers had been doing since Christian Huygens in Newton’s time.¹

汉密尔顿在对光波穿过各种形状的晶体时的行为方式进行新分析时发现，当这些波以特定方向进入某种类型的晶体（称为“双轴”）时，它们应该被折射成一个射线锥，而不是你在普通棱镜中看到的熟悉的带状。如今，激光束的“锥形折射”正在找到各种实际用途，包括作为操纵化学分子或血细胞的光镊，以及自由空间光通信^。2

In his new analysis of the way light waves should behave when they pass through crystals of various shapes, Hamilton found that when these waves passed into a certain type of crystal (called “biaxial”) in a certain direction, they should be refracted into a cone of rays rather than the familiar band you see with an ordinary prism. Today, the “conical refraction” of laser beams is finding various practical uses, including as optical tweezers for manipulating chemical molecules or blood cells, and in free-space optical communication.²

然而，早在 1832 年初，还没有人见过圆锥折射这样的现象（更不用说激光了）。于是，汉密尔顿请他的实验学家朋友汉弗莱·劳埃德看看他能否找到这种新的折射——当他找到这种折射时，汉密尔顿一举成名，因为这可能是数学第一次被用来预测物理现象的存在，而不是解释已知的现象。随后出现了更多这样的突破：十年后，海王星的存在被预测出来，然后是一系列的预测，从无线电波和辐射压力到光子和E = mc ²到希格斯玻色子、引力波，等等。

Back in early 1832, however, no one had ever seen such a phenomenon as conical refraction (let alone lasers). So Hamilton asked his experimentalist friend Humphrey Lloyd to see if he could find this new kind of refraction—and when he did, Hamilton became a sensation, for it was perhaps the first time that mathematics had been used to predict the very existence of a physical phenomenon, rather than explaining what was already known. Many more such breakthroughs were to come: the prediction of the existence of Neptune would follow a decade later, and then a host of predictions, from radio waves and radiation pressure to photons and E = mc² to the Higgs boson, gravitational waves, and much, much more.

• • •

• • •

汉密尔顿十几岁时就发表了第一篇关于光学的数学论文，而当他关于圆锥折射的数学预测得到验证时，他才 27 岁。而他做出这一惊人预测所需的工具之一就是微积分。

Hamilton had presented his first mathematical paper on optics when he was still a teenager, and he was still only twenty-seven when his mathematical prediction of conical refraction was verified. And among the tools he’d needed to make his remarkable prediction was calculus.

微积分有两个分支，微分和积分。微分学本质上是关于变化率的，比如速度，即距离随时间的变化率。在关于折射的研究中，汉密尔顿使用导数来显示某些量（与光程长度和波速有关）在晶体表面的变化，但所有形式的平滑变化现象可以用微积分来建模：加热和冷却、生物和原子的生长和衰变、各种波、生态系统、热和地形梯度、数学函数的梯度（也可用于解决优化问题）、金融趋势等等。

Calculus has two branches, differential and integral. Differential calculus is essentially about rates of change—as in speed, the rate of change of distance with time. In his work on refraction, Hamilton used derivatives to show how certain quantities (related to optical path length and wave speed) changed at the faces of the crystal, but all manner of smoothly changing phenomena can be modeled by differential calculus: heating and cooling, growth and decay both biological and atomic, waves of various kinds, ecological systems, thermal and topographical gradients, the gradients of mathematical functions (which can be used also to solve optimisation problems), financial trends, and so on.

至于积分，它是一种计算工具，用于解决古代数学家所面临的问题，例如计算田地和建筑平面图的面积或蓄水池和运河的体积。在当今的高科技社会中，还有更多这样的应用——例如计算飞机机翼升力面的面积；确定制造产品（从计算机和车身到桥梁和摩天大楼）所需材料的平方英尺数；确定建筑所用材料的重量（通过密度和体积），以确保结构具有足够的承重能力；在手术期间监测患者的血液量……我还可以继续列举下去，但您已经看到了范围。

As for integral calculus, it developed as a calculation tool to solve just the kinds of problems that ancient mathematicians had grappled with, when they wanted to calculate such things as the areas of fields and floor plans or the volumes of cisterns and canals. In today’s high-tech society there are many more such applications—calculating the area of the lifting surface of an airplane’s wing, for example; determining the square footage of material needed for manufacturing products, from computers and car bodies to bridges and skyscrapers; finding the weight (via the density and volume) of materials used in construction, to ensure the structure is sufficiently load-bearing; monitoring the volume of a patient’s blood during surgery … I could go on, but you can see the scope.

您可能最初是通过近似计算曲线y = f ( x ) 下的面积来学习积分的，其中f ( x ) 表示描述您感兴趣的曲线的x函数；例如，对于以原点为中心和基础的抛物线，则有f ( x )= x2。您可以使用许多细长的矩形来近似该面积，每个矩形的面积为^ydx，如图 2.1所示；添加的矩形越多，每个矩形就越细，近似值就越好。这是一种直观的几何方法，但是使用积分算法的代数公式，您可以“相加”无数个这些无穷小矩形来获得精确的面积。您可以做一些类似的事情来求体积，将无数个圆柱切片或壳相加，每个圆柱切片或壳的体积都是无穷小的。

You likely first learned integral calculus by approximating the area under a curve y = f(x), where f(x) represents the function of x that describes the curve you’re interested in; for instance, for a parabola centred and based at the origin, you’d have f(x) = x². You’d approximate this area with a lot of skinny rectangles, each of area ydx, as in figure 2.1; the more rectangles you add, the thinner each one becomes and the better the approximation. It’s an intuitive geometrical approach, but with algebraic formulae for the algorithms of integral calculus, you can “add up” an infinite number of these infinitesimal rectangles to get the exact area. You do something similar to find volumes, adding up an infinite number of cylindrical slices or shells, each of infinitesimal volume.

“加一个无穷数……”这句话在今天听起来很容易，但能够操纵无穷数确实是一件了不起的事情。古代数学家完全回避了这个概念——就像他们对待我们之前遇到的虚数一样。

Saying “adding up an infinite number …” trips easily off the tongue today, but it’s really quite a marvel to be able to manipulate infinity. Ancient mathematicians had shied away from the concept altogether—just as they’d done with those imaginary numbers we met earlier.

例如，今天如果你想严格推导圆的面积（πr2）或周长（2πr）的公式，^你可以对其积分代数方程，如图 2.3所示。相比之下，三千五百年前，埃及抄写员阿赫梅斯在奇迹般保存至今的纸莎草卷轴上记录了一个用八边形近似圆形的简单几何结构。由此产生的面积计算结果令人惊讶地接近我们现在称为 π 的常数，如图2.2所示。要找到近似多边形的周长，阿赫梅斯只需将八边形的边加起来——使用毕达哥拉斯定理来找到角的长度^。3

Today, for instance, if you wanted to derive rigorously the formula for the area (πr²) or circumference (2πr) of a circle, you’d integrate its algebraic equation, as we’ll see in figure 2.3. Three and a half thousand years ago, by contrast, the Egyptian scribe Ahmes recorded—on a papyrus scroll that miraculously still survives—a simple geometric construction that approximates a circle with an octagon. The resulting area computation gives a surprisingly good approximation for the constant we now call π, as you can see in figure 2.2. To find the circumference of the approximating polygon, Ahmes just had to add up the sides of the octagon—using Pythagoras’s theorem to find the lengths of the corners.³

在阿赫梅斯记录下他那个时代的数学一千年后，希腊数学家们发明了“穷举法”（后来被描述为穷举法），其中圆被多边形逐渐逼近，边数逐渐增加，直到达到所需的精度。这是严谨性方面的一次出色尝试——许多古代数学家对证明和计算的想法都很感兴趣——锡拉丘兹的阿基米德就是其中一位，他很好地利用了它。为了找到圆的周长，他从一个内接正六边形开始，然后将边数翻倍，比较两个连续多边形的周长。他不断地翻倍和比较，就这样，他找到了一种算法，用于添加和比较多达九十六条边的多边形的线段！这无疑是一项令人印象深刻的计算壮举，如果你想象自己试着画出它，你会发现九十六条边的多边形是一个漂亮的对圆的精确近似——阿基米德能够将（我们所说的）π 的值估计为介于$3\frac{10}{71}$和$3\frac{10}{70}$上限是我们很多人在学校学到的价值观：$3\frac{1}{7}$或 22/7（与现代值 3.14159 相比，为 3.14286，均四舍五入到小数点后五位）。

A thousand years after Ahmes recorded the maths of his day, Greekspeaking mathematicians developed “the method of exhaustion” (as it was later described), in which circles were successively approximated by polygons with an increasing number of sides, up to a desired accuracy. It was a brilliant attempt at rigour—many ancient mathematicians were interested in the idea of proof as well as in computation—and Archimedes of Syracuse, for one, made fine use of it. To find the circumference of a circle, he started with an inscribed regular hexagon, and then doubled the number of sides, comparing the perimeters of the two successive polygons. He kept on doubling and comparing, and, in this way, he found an algorithm for adding and comparing the segments of polygons with up to ninety-six sides! It was certainly an impressive feat of computation, and if you imagine trying to draw it, you’ll see that a ninety-six-sided polygon is a pretty fine approximation to a circle—good enough that Archimedes was able to estimate the value of (what we call) π as somewhere between $3\frac{10}{71}$ and $3\frac{10}{70}$. The upper limit is the value that many of us learned in school: $3\frac{1}{7}$ or 22/7 (which is 3.14286 compared with the modern value of 3.14159, both rounded to five decimal places).

不过，将九十六条边相加并不等同于将多边形变成真正的圆形所需的无数条边相加。相比之下，在微积分中，曲线的长度是通过对曲线的微小部分的长度代数公式进行积分来求得的，用ds表示。它是阿基米德九十六条边之一的无穷小版本，其长度是通过类比毕达哥拉斯定理得出的。对于图 2.3b中的示例，您可以看到，即使与 Ahmes 的方法也有祖先的相似之处。

Still, adding up ninety-six sides is not the same as adding up the infinite number of them required to turn the polygon into a true circle. With calculus, by contrast, lengths of curves are found by integrating the algebraic formula for the length of a minute segment of the curve, denoted by ds. It’s an infinitesimal version of one of Archimedes’s ninety-six sides, and its length is found by analogy with Pythagoras’s theorem. For the example in figure 2.3b, you can see there’s an ancestral similarity even with Ahmes’s approach.

将八边形的边相加（对角线使用毕达哥拉斯定理），并将其与半径为的圆的周长的现代公式进行比较$\frac{9}{2}$，你会得到埃及人对 π 的近似值为 3.2 — 仍然与他们的 3.16 大致相同。

Adding up the sides of the octagon (using Pythagoras’s theorem for the corners) and comparing this with the modern formula for the circumference of a circle of radius $\frac{9}{2}$, you get the Egyptian approximation of π to be 3.2—still in the same ballpark as their 3.16.

如果阿基米德没有在公元前 212 年罗马人围攻锡拉库萨时被杀，他可能更接近发明微积分。他当时大约 75 岁，杀死他的士兵显然不尊重长辈或数学——据说阿基米德沉浸在计算中，没有听到士兵的命令。两千年后，在另一场暴力动乱——法国大革命的黑暗日子里，一位了不起的年轻女孩读到了这个故事。她对阿基米德的迷恋感到非常兴奋，决定学习显然令人兴奋的数学学科——但她必须自学，因为当时大多数女孩都无法接受正规教育。而且她必须秘密学习，因为她的父母和几乎所有人一样，认为由于女性通常比男性体格更小、更弱，认真学习会损害她们的健康，阻止她们生育，甚至让她们发疯。她就是索菲·热尔曼，她成为二十世纪前最优秀的女数学家之一。她在最重要的工作中使用了微积分，开创了振动表面的数学理论。

Archimedes might have come even closer to inventing calculus if he hadn’t been killed during the Roman siege of Syracuse in 212 BCE. He was about seventy-five years old, and the soldier who killed him apparently had no respect for elders or for mathematics—Archimedes was reputedly so immersed in his calculations that he hadn’t heard the soldier’s orders. Two thousand years later, during the dark days of another violent upheaval, the French Revolution, a remarkable young girl read this story. She was so excited by Archimedes’s thrall that she decided to learn the evidently thrilling subject of mathematics—but she had to teach herself because most girls were denied a proper education at that time. And she had to do it in secret, because her parents believed, like almost everyone else, that because women were generally physically smaller and weaker than men, serious study would damage their health, stopping them from having babies or even driving them mad. She was Sophie Germain, and she became one of the best female mathematicians before the twentieth century. She used differential calculus in her most important work, pioneering the mathematical theory of vibrating surfaces.

尽管阿基米德的死很悲惨，但如果他最重要的手稿没有被中世纪的抄写员部分洗掉并改写，微积分也许会更早出现。直到 1906 年，手稿才被恢复，表明阿基米德确实发现了一些非凡的成果。仅凭具体几何和机械类比，他就设法找到了球体和曲面容器的体积，所用方法预示了如今本科积分微积分课上教授的“旋转体”的体积。但这项天才之作、这一信息宝藏却失传已久，以至于早期现代数学家不得不自己重新发现其中的思想。

Even despite Archimedes’s tragic death, calculus might have arrived sooner if his most important manuscript on the subject hadn’t been partly washed off and written over by a medieval scribe. It was recovered only in 1906, and it showed that Archimedes had found some truly extraordinary results. Working just with concrete geometry and mechanical analogies, he’d managed to find the volumes of spheres and curved-shaped containers, using methods that prefigure the volumes of “solids of revolution” taught today in undergrad integral calculus classes. But this work of genius, this treasure of information, was lost for so long that early modern mathematicians had had to rediscover its ideas for themselves.

牛顿天才的标志之一是，他对数学和物理学的许多分支做出了杰出的贡献：在现代从广义上讲，他是一位实验物理学家，尤其擅长光的研究；他是一位理论物理学家，尤其擅长运动和引力理论的研究；他是一位应用数学家（就他本人以及本文中的其他几位作者以及当今的许多研究人员而言，当数学应用于自然物理过程时，应用数学与理论物理学是相交叉的）；他是一位纯数学家，他发明了新的数学方法、概念和证明，注重严谨性和美感。这些“纯”技术往往为应用数学家和理论物理学家后来的实际应用奠定了基础，微积分就是一个典型的例子。

One of the hallmarks of Newton’s genius is that he was an outstanding contributor to so many branches of mathematics and physics: in modern terms, he was an experimental physicist, especially on the subject of light; he was a theoretical physicist, especially with his theories of motion and gravity; he was an applied mathematician—a category that in his case, and for several others in this story as well as for many researchers today, intersects with theoretical physics when mathematics is applied to natural physical processes—and he was a pure mathematician, one who invents new mathematical methods, concepts, and proofs, with a focus on rigour and beauty. These “pure” techniques often underpin the later real-world applications of the applied mathematicians and theoretical physicists— and calculus is a classic example.

牛顿首次发现了微分和积分这两种微积分的通用符号表示算法，清楚地表明了它们之间的联系。也就是说，他证明了积分是微分的逆运算。戈特弗里德·莱布尼茨也意识到了这一点，因为微积分当时“尚在空中”，这要归功于哈里奥特和伽利略、皮埃尔·德·费马（费马最后定理的提出者）、艾萨克·巴罗（牛顿在剑桥的导师）和约翰·沃利斯等先行者。正是通过他们的工作以及他们所借鉴的工作，从阿基米德和欧几里得等古人到阿布·阿里·伊本·海赛姆和尼科尔·奥雷斯姆等中世纪学者，莱布尼茨和牛顿才得以各自独立地将微积分形成其第一个明确形式。

Newton was the first to find general, symbolically represented algorithms for the two types of calculus, differential and integral, clearly showing the connection between them. That is, he showed that integration is the inverse operation of differentiation. Gottfried Leibniz got the idea, too, for calculus was “in the air,” thanks to forerunners such as Harriot and Galileo, Pierre de Fermat (of Fermat’s last theorem fame), Isaac Barrow (Newton’s mentor at Cambridge), and John Wallis. It is through their works—and those they built on, from ancients such as Archimedes and Euclid to medieval scholars such as Abu Ali Ibn al-Haytham and Nicole Oresme—that Leibniz and Newton were able, each independently, to put calculus into its first definitive form.

这两位创始人截然不同。牛顿对他的发现讳莫如深，因为他无法忍受被批评。他对很多事情都很奇怪，尽管我确实同情他。他在微积分、光学和引力理论方面取得了最初的突破，当时他只有二十出头——藏在母亲的农场里，在 1665-67 年的瘟疫年代与世隔绝。这样的成就很难不让人敬畏，也很难让人不为内心受惊的孩子感到难过。在他出生之前，他的父亲就在奥利弗·克伦威尔的议会党和查理一世的保皇党之间的内战中去世了。对于他的母亲汉娜来说，那一定是一段可怕的时光，她的丈夫死了，战斗如此激烈——距离她的农场只有附近的村庄。于是她嫁给了富裕的牧师巴纳巴斯·史密斯，并搬到了另一个城镇；不幸的是——显然是在史密斯的坚持下——她把年幼的艾萨克和她的母亲留在了那里。众所周知，他希望自己能把史密斯的房子烧毁，把史密斯和他母亲都烧了，但他似乎唯一表面上的破坏性行为就是偶尔画一些艺术涂鸦，或者偶尔在晚上放爆炸的风筝恶作剧，把迷信的村民吓得魂飞胆丧。直到史密斯去世后，汉娜才带着三个不受欢迎的同父异母兄弟姐妹回到艾萨克身边，当时艾萨克大约十岁^。4

These two founders could not have been more different. Newton was rather secretive about his discoveries because he couldn’t stand the thought of being criticised. He was rather weird about a lot of things, although I do have some sympathy for him. He made his initial breakthroughs with calculus, optics, and the theory of gravity when he was still only in his early twenties—tucked away at his mother’s farm, isolating during the plague years 1665–67. It’s hard not to be awed by such an achievement. And it’s hard not to feel for the frightened child inside. Before he was born, his father had died in the Civil War between Oliver Cromwell’s Parliamentarians and King Charles I’s Royalists. It must have been a terrifying time for his mother, Hannah, with her husband dead and the fighting so fierce—it came as close to her farm as the nearby village. And so she married Barnabas Smith, a well-off rector, and moved to another town; unfortunately—and apparently at Smith’s insistence—she left young Isaac behind with her mother. He famously wished he could have burned Smith’s house down with his mother and Smith inside, but it seems the only outwardly destructive behavior he engaged in was the odd bit of artistic graffiti or the occasional night-time pranks with exploding kites that frightened the wits out of superstitious villagers. Only when Smith died did Hannah return to Isaac, then aged about ten, with three unwelcome half-siblings in tow.⁴

莱布尼茨比牛顿阳光得多。莱布尼茨学者菲利普·维纳说，莱布尼茨是一位真正的文艺复兴人，他“是一位律师、科学家、发明家、外交官、诗人、语言学家、逻辑学家，也是一位虔诚地捍卫理性培养的哲学家，认为理性培养是人类进步的光辉希望” ^。5

Leibniz was a much sunnier person than Newton. A true Renaissance man, he was “a lawyer, scientist, inventor, diplomat, poet, philologist, logician, and a philosopher who religiously defended the cultivation of reason as the radiant hope of human progress,” to quote Leibniz scholar Philip Wiener.⁵

莱布尼茨六岁时，身为哲学教授的父亲去世。莱布尼茨的外祖父也曾是法学教授，年轻的戈特弗里德后来不仅获得了法学学位，还获得了博士学位。相比之下，牛顿的父亲是一位目不识丁的农民，他的母亲并不理解他对知识的热爱和天赋：她在他十七岁时就让他辍学，直到他把农场经营得一团糟，母亲最终接受了校长和她在剑桥大学毕业的哥哥的建议，允许艾萨克为上大学做准备。但他必须自费完成剑桥学业，为一位富有的学生做“sizar”或仆人。对他和我们来说幸运的是，他很快就成为了那里的教授，可以自由地做出他惊人的发现。

When Leibniz was six, his father, a philosophy professor, died. His mother’s father had also been a professor—of law, in his case, the field in which young Gottfried later earned not just a degree but also a doctorate. By contrast, Newton’s father had been an illiterate farmer, and his mother had not understood his intellectual passion and talent: she took him out of school when he was seventeen, until he made such a mess of running the farm that she finally accepted the advice of both the schoolmaster and her Cambridge-educated brother, and allowed Isaac to prepare for university. But he had to pay his own way through Cambridge, working for a wealthy student as a “sizar” or servant. Fortunately for him and for us, he soon became a professor there, and was free to make his amazing discoveries.

莱布尼茨在学生时代没有这样的烦恼，毕业后他过着相当光鲜的生活。他为他的王室赞助人在欧洲执行外交任务，一路上与科学家和哲学家交谈，建立了一个由大约六百名通信者组成的庞大网络——他是一位多产的书信作家，对世界和平、哲学和科学有着远大的想法。鉴于他广泛的兴趣和职业职责，他表现出非凡的数学天赋。毫不奇怪，学术上的牛顿是更优秀的数学家和实验物理学家，但兼收并蓄的莱布尼茨仍然做出了重大发现——他是更好的符号制造者。

Leibniz had no such worries as a student, and after graduating he led a rather glamorous life. Traveling through Europe on diplomatic missions for his princely patrons and conversing with scientists and philosophers along the way, he built up a huge network of some six hundred correspondents—he was a prolific letter writer, with big ideas about universal peace, philosophy, and science. Given all his wide-ranging interests and his professional duties, he showed an extraordinary talent for mathematics. Not surprisingly, the academic Newton was the better mathematician and experimental physicist, but the eclectic Leibniz made significant discoveries nonetheless—and he was the better symbol-maker.

在上一章中，我强调了代数从文字到符号的缓慢发展，但在现代微积分的早期——在维埃特、哈里奥特和笛卡尔引入真正的代数符号主义半个世纪之后——情况发生了逆转：微积分符号代表了当时没有人能够用文字充分解释的想法。这就是符号思维的特殊力量：它可以带你进入最初超出普通理解的新领域。

In the previous chapter, I highlighted algebra’s slow progress from words to symbols, but in the early days of modern calculus—half a century after Viète, Harriot, and Descartes introduced true algebraic symbolism— the situation was reversed: calculus symbols stood for ideas that no one at the time could adequately explain in words. This is the special power of symbolic thinking: it can take you into new places that are, at first, beyond ordinary understanding.

例如，尽管我一直在轻松地谈论“无穷大”和“无穷小”，但它们是很难有意义地定义的概念。现代词典对“无穷小”的定义是“极其小”，但这对数学家没有多大帮助。他们需要更精确的定义，尽管在这种情况下，在牛顿-莱布尼茨微积分诞生两百年后才找到这样的定义（通过极限、连续性和函数理论）。当你考虑到要找到一个运动物体在任何给定时间的速度，你必须将它当时的位置与一瞬间之后的位置进行比较，然后将差值除以时间的瞬间时，你就会明白其中的困难。但你所说的“瞬间”是什么意思呢？你如何定义位置的增量变化？显然，它们都是非常小的量，但有多小呢？

For instance, although I’ve been talking easily about “infinity” and “infinitesimal” quantities, they are concepts that are very difficult to define meaningfully. A modern dictionary definition of “infinitesimal” is “extremely small,” but that’s not much help to mathematicians. They need something more precise, although in this case it took two hundred years after the birth of Newtonian-Leibnizian calculus to find such a definition (via the theory of limits, continuity, and functions). You can see the difficulty when you consider that to find a moving object’s speed at any given time, you have to compare its position at that time with its position an instant later, and then divide the difference by the instant of time. But what do you mean by an “instant”? And how do you define the incremental change in position? Obviously, they are both very small quantities, but how small?

牛顿将这些无穷小的变化称为“矩”，用符号 ο 表示（希腊字母 omicron 表示“不完全为零”）。莱布尼茨将它们称为“差”或“微分”，并将它们表示为dt、dx等。对于距离x相对于时间的变化率，牛顿使用一个点，如ẋ。另一方面，莱布尼茨则用“比率”来表示，例如x相对于t的变化率，以及y相对于$十-\text{作为}\frac{d十}{d吨}，\frac{d是}{d十}$等等。实际上，莱布尼茨主要使用dy : dx符号，其中冒号表示比率；是他的追随者——尤其是约翰·伯努利（他的名字通常以法语方式写成 Jean，有时也写成 John）——将这个比率推广为$\frac{d是}{d十}$。你可以看到谁的符号最终胜出——尽管点符号仍然用于区分时间，这对于研究运动、波、场以及随时间变化的其他各种量非常重要。

Newton called these infinitesimal changes “moments,” denoting them by the symbol ο (a “not quite zero” represented by the Greek letter omicron). Leibniz called them “differences” or “differentials,” and denoted them dt, dx, and so on. For the rate of change of a distance x with respect to time, Newton used a dot, as in ẋ. Leibniz, on the other hand, wrote his “ratios”—his rates of change of x with respect to t, for instance, and of y with respect to $x-\text{as}\frac{dx}{dt},\frac{dy}{dx}$, and so on. Actually, Leibniz mostly used the notation dy:dx, where the colon symbolised a ratio; it was his followers—especially Johann Bernoulli, whose first name is commonly written in the French way, Jean, and occasionally Anglicised as John—who popularised this ratio as $\frac{dy}{dx}$. You can see whose symbols eventually won the day—although the dot notation is still used for differentiation with respect to time, which is important in the study of motion, waves, fields, and various other quantities that change over time.

牛顿和莱布尼茨都意识到了“无穷小”一词背后的概念问题。莱布尼茨尝试通过改进古老的穷举法来解释它，而牛顿则对我们现在所说的极限做出了合理的早期定义。（您可以在尾注⁶中看到他们的尝试，以及与之相对的现代定义，我在下面的图 2.4中勾勒出了极限的概念。）然而，在计算中，莱布尼茨和牛顿经常做任何精明的学生都会做的事情：忽略这些微小的增量和瞬间，因为它们太小而不必担心——有点像你花 124.99 美元买一件商品，不用担心 125.00 美元的零钱。它在实践中很有效，只是像一分钱一样，“瞬间”的时间实际上并不为零。如果你把它当成零，当你想除以“瞬间”时，就会遇到理论问题：例如，要找到瞬时速度，你就得做不可能的事，除以零。更糟糕的是，距离的微小变化也近似为零，所以你会试图计算 0/0。

Both Newton and Leibniz were aware of the conceptual problem underlying the word “infinitesimal.” Leibniz had a stab at explaining it by refining the ancient method of exhaustion, and Newton made a reasonable fist of an early definition of what we now call limits. (You can see their attempts, and by comparison a modern definition, in the endnote,⁶ and I’ve sketched the limit idea in fig. 2.4 below.) In their calculations, though, Leibniz and Newton often did what any canny student would do: ignore these tiny increments and instants as too small to worry about—a little like when you buy an item for $124.99 and don’t worry about the change out of $125.00. It works well in practice, except that like a cent, an “instant” of time is not actually zero. If you act as though it is zero, you run into theoretical problems when you want to divide by an “instant”: to find an instantaneous speed, for example, you’d be doing the impossible and dividing by zero. Worse, the tiny change in distance is approximately zero, too, so you’d be trying to calculate 0/0.

阿基米德发现，将无穷小量相加已经够难了，更不用说将它们相除求导数了——早在公元前 450 年，传奇人物芝诺就在其著名悖论中强调了这一困难。例如，他说，要从 A 跑到 B，首先你必须跑到 B 的一半路程；但在跑到一半之前，你必须跑四分之一，以此类推，直到无限。这意味着你永远无法真正开始。当然，每个人都知道你可以从 A 跑到 B。芝诺似乎在这里表明——至少从现代的角度来看——运动的概念是多么难以定义，因为它的距离是无穷小的，时间是微小的“瞬间”。当然，芝诺本人可能有神秘的目的，而不是数学的目的，所以他的悖论——就像他同时代的佛教徒的教义一样——可能是为了表明时间和空间是幻觉。

It was hard enough trying to add infinitesimal quantities, as Archimedes found, let alone divide them to find derivatives—and as long ago as 450 BCE, the legendary Zeno had highlighted the difficulty in his famous paradoxes. For instance, he said, to run from A to B, first you have to run halfway to B; but before you reach halfway, you have to run a quarter of the way, and so on, indefinitely. Which means you can never really get started. Of course, everyone knows that you can run from A to B. What Zeno seemed to be showing here—from a modern perspective, at least—was just how difficult it is to define the idea of motion, with its infinitesimal increments in distance and its tiny “instants” of time. Of course, Zeno himself may have had a mystical rather than a mathematical purpose, so it’s possible that his paradoxes—like the teachings of his Buddhist contemporaries— were intended to suggest that time and space are illusions.

尽管极限和收敛的概念直到现代才被严格证明，但古代和中世纪的数学家已经使用巧妙的几何结构来求有限个级数的和。他们用文字写下级数和算法——哈里奥特似乎是第一个像我们今天一样用数字和字母符号重写级数的人，他在 1600 年左右做到了这一点。特别是在一次漫长而精湛的计算中，他使用无限几何级数和直观的极限论证来找到一艘船沿着固定罗盘方位在弯曲的地球上绘制的螺旋线长度的代数表达式。（他的赞助人沃尔特·雷利爵士希望获得最好的科学专业知识来帮助提高航行的安全性和效率，因为他正在计划从英国到美国的未知海洋的航行。）哈里奥特首先将螺旋线分解成小段，就像阿基米德的圆一样，但他假设它们的数量是无限的，并且大小不断减小。他花了数十页纸和数月的时间才完成这项工作——这是除早期对圆周长的估计之外，第一个现存的精确曲线长度推导。如今，积分微积分用于求出这种长度，如图2.3b所示。不过，哈里奥特会很高兴知道他的代数符号方法直接影响了英国微积分先驱约翰·沃利斯，而沃利斯又影响了年轻的牛顿^。7

Although the ideas of limits and convergence weren’t proven rigorously until modern times, ancient and medieval mathematicians had used ingenious geometric constructions to find the sums of a limited number of series. They wrote both their series and their algorithms in words— Harriot seems to have been the first to rewrite series in terms of number and letter symbols as we do today, which he did sometime around 1600. In particular, in a long and virtuoso calculation, he used an infinite geometric series and an intuitive limit argument to find an algebraic expression for the length of the spiral traced over the curved Earth by a ship following a fixed compass bearing. (His patron, Sir Walter Ralegh, had wanted the best scientific expertise to help make navigation safer and more efficient, for he was planning voyages from England to America across an uncharted ocean.) Harriot began by breaking up the spiral into small segments like Archimedes’s circle, except he assumed an infinite number of them, of ever-decreasing size. He took dozens of pages and many months to conclude this work—the first extant exact derivation of the length of a curved line, other than those early estimates of the circumference of a circle. Today, integral calculus is used to find such lengths, as figure 2.3b showed. Still, Harriot would have been chuffed to know that his symbolic approach to algebra directly influenced the English calculus pioneer John Wallis, and Wallis influenced the young Newton.⁷

图 2.3b和相关方框还表明，要使用微积分来计算曲线的长度，你需要一个曲线公式——这就是解析几何的领域。哈里奥特也开创了这一领域，但真正将“解析”几何或代数几何发扬光大的人是笛卡尔，为了纪念他，我们将矩形x、y、z坐标系称为“笛卡尔坐标系”。即便如此，笛卡尔也几乎没能意识到用以这些坐标写出的代数方程来表示几何曲线的非凡威力（例如上图 2.3中圆的代数方程为x ² + y ² = r ²）。他的主要目的是表明代数和几何结合起来可以使解决问题变得更容易：代数符号使大脑不必将复杂的几何构造形象化（例如卡尔达诺的完整立方体），而几何则为代数运算赋予具体的含义。二十年后，沃利斯对此事不那么公正，他绝对更喜欢代数而不是几何。

Figure 2.3b and the related box also showed that to use calculus to find the length of a curved line, you need a formula for your curve—and this is the province of “analytic geometry.” Harriot had made a start at this, too, but it was Descartes who really put “analytic” or algebraic geometry on the map, and we call the rectangular x, y, z coordinate system “Cartesian” in his honour. Even so, Descartes barely glimpsed the remarkable power of being able to represent geometric curves by algebraic equations written in terms of these coordinates (such as x² + y² = r² for the circle in fig. 2.3 above). His main purpose had been to show that algebra and geometry together make problem-solving easier: algebraic symbolism frees the mind from having to visualise complicated geometric constructions (such as Cardano’s completed cube), while geometry gives concrete meaning to algebraic operations. Twenty years later, Wallis was less evenhanded about the matter, definitely favouring algebra over geometry.

沃利斯之所以更喜欢英国人而不是法国人，是因为他对于法国人（以费马和其他几个人为代表）对他早期著作的反应感到恼火。他还发誓，而且他并不是^唯一一个发誓笛卡尔抄袭了哈里奥特的《实践》一书的人。8无论如何，沃利斯在 1655 年出版的《无穷算术》一书中向代数微积分迈出了重要一步。他甚至创造了一个表示无穷的符号——就是我们今天使用的 ∞。罗马人曾偶尔使用这个符号来表示 1,000，它也时不时地出现在各种语境中，包括埃及图像的一个版本，图中一条蛇在吃着自己的尾巴，象征着生死无尽的循环。没有人知道沃利斯为什么用它来表示无穷；也许他对它的古老神秘含义印象深刻，尽管作为加尔文教徒，他选择它更有可能只是因为它代表了一条永无止境的曲线。

Wallis also favoured the English over the French, because he’d been miffed by the French response—in the persons of Fermat and several others—to his own early work. He also swore, and he wasn’t entirely alone in this, that Descartes had cribbed from Harriot’s book Praxis.⁸ Anyway, Wallis took a significant step toward an algebraic calculus in his 1655 book Arithmetica Infinitorum. He’d even created a symbol for infinity—the same one we use today, ∞. This symbol had been used occasionally by the Romans to represent 1,000, and it also cropped up every now and then in various contexts, including a version of the Egyptian image of a snake eating its own tail and symbolising the endless cycle of birth and death. No one knows why Wallis used it for infinity; perhaps he was impressed with its ancient mystical connotations, although as a Calvinist he more likely chose it simply because it represents a never-ending curve.

沃利斯的故事让我们得以一窥当时的数学和政治背景，就在牛顿崭露头角之前。他是肯特郡一位乡村牧师的儿子，幸运的是，他的家人认可并支持他的学术倾向。他最终去了剑桥，尽管在那里他没学到多少数学。直到圣诞节假期回家时，他才意识到这门学科的存在，当时他注意到弟弟正在学习算术，为做职业做准备。他很感兴趣，要求上一堂课。正如他后来所说，

Wallis’s story gives a glimpse into the mathematical and political context at that time, just before Newton burst onto the scene. He was the son of a village rector in Kent, and fortunately his family recognised and supported his academic bent. He eventually went to Cambridge, although he didn’t learn much maths there. He’d only realised the subject existed when he came home for the Christmas break and noticed his younger brother studying arithmetic in preparation for a trade. Intrigued, he’d asked for a lesson. As he put it later,

那时，数学并不被视为学术知识，而是商人、海员、木匠、土地测量员或类似人员的工作……当时我们学院（剑桥大学伊曼纽尔分校）有 200 多名学生，我不知道有哪两个人比我更懂数学，而我懂的数学知识非常少；直到我被任命为数学教授之前，我从未认真学习过数学（除了作为一种愉快的消遣）^。9

Mathematicks were not, at that time, looked upon as Academical Learning, but the business of Traders, Merchants, Sea-men, Carpenters, land-measurers, or the like.... Of more than 200 at that time in our College [Emmanuel, Cambridge], I do not know of any two that had more Mathematicks than myself, which was but very little; having never made it my serious studie (otherwise than as a pleasant diversion) till some little time before I was designed for a Professor in it.⁹

人们常说，学习某种东西的最好方法就是教它！而沃利斯肯定学得很好，因为他成为了牛顿之前那一代最优秀的数学家之一。然而，这真是一段不平凡的旅程。首先，他获得了神学硕士学位。然后，他获得了剑桥大学皇后学院的研究员职位，但是这份职位并没有维持多久：因为当时研究员必须保持未婚状态，因此在 1645 年，新婚的沃利斯成为了一名牧师而不是学者。然后，出乎意料的是，他发现自己有破译密码的天赋。当时南北战争正在激烈进行，一位牧师同事向他展示了一条被截获的密码信息，半开玩笑地问他能否理解。令他自己惊讶的是，沃利斯在几个小时内就破译了这条信息。这是他人生的转折点，因为当国王于 1649 年被处决时，如果你能证明自己对胜利者的忠诚，那将非常有利。沃利斯为议会党破解了许多密码，还展示了他的数学技能——当牛津大学的萨维尔几何学教授因保皇党而被解雇时，沃利斯被任命。（尽管沃利斯支持议会党，但他曾公开反对处决国王，并相当勇敢地签署了一份抗议文件。因此，当君主制后来恢复时，他仍然支持。此外，他更喜欢数学而不是政治，而且正如他告诉保皇党朋友的那样，他解开的大多数信息对议会党来说根本没多大用处。）¹⁰

They do say that the best way to learn something is to teach it! And Wallis must have learned it well, for he became one of the best mathematicians in the generation before Newton. It had been quite a journey, though. First, he graduated with a master of arts in divinity. Then he took up a fellowship at Queen’s College, Cambridge, but it was short-lived: fellows had to remain unmarried in those days, so in 1645, the newly wedded Wallis became a minister instead of an academic. And then, quite out of the blue, he discovered he had a knack for code breaking. The Civil War was raging, and a fellow clergyman had shown him an intercepted message in cipher, asking half-jokingly if he could make sense of it. To his own amazement, Wallis was able to decipher the message within a few hours. It was a turning point in his life, for when the king was executed in 1649, it was very handy if you’d proved your loyalty to the victors. Wallis had broken a number of codes for the Parliamentarians, showing his mathematical skills into the bargain—and when Oxford’s Savilian Professor of Geometry was sacked for having been a Royalist, Wallis was appointed instead. (Although he supported the Parliamentarians, Wallis had spoken out against the execution of the king, and had signed, rather bravely, a document protesting it. So when the monarchy was later restored, he remained in favour. Besides, he preferred mathematics to politics, and as he told his Royalist friends, most of the messages he’d unlocked hadn’t been of all that much use to the Parliamentarians anyway.)¹⁰

沃利斯在牛津大学任职仅仅六年后就发表了他的代数著作《无限算术》。当时，大多数数学家们仍然喜欢使用有形的几何思想来开发处理无穷和和无穷小增量的方法——托马斯·霍布斯在这方面尤其直言不讳。他今天的名声来自于他的政治哲学——以及他得出的严肃结论：没有国家的保护，生活是“肮脏、野蛮和短暂的”。但他也涉足数学，他表达了这种支持几何学的情绪，称沃利斯的《无限算术》是一本“肮脏的书”和“一堆符号”。事实上，他谴责“所有将代数应用于几何学的人”。显然，他不是一个开朗的人。

It was just six years after his appointment at Oxford when Wallis published his algebraic Arithmetica Infinitorum. At the time, most mathematicians still preferred using tangible geometric ideas to develop ways of handling infinite sums and infinitesimal increments—and Thomas Hobbes was particularly vociferous on the subject. His fame today rests on his political philosophy—and his dour conclusion that life is “nasty, brutish, and short” without the protection of a state. But he also dabbled in mathematics, and he expressed this pro-geometrical sentiment by pronouncing Wallis’s Arithmetica Infinitorum a “scurvy book” and “a scab of symbols.” In fact, he denounced “the whole herd of them who apply their algebra to geometry.” He evidently was not a sunny person.

尽管霍布斯和他的几何学家同行们对此持反对态度，但十年后，沃利斯的著作让年轻的牛顿首次踏上了学习微积分的道路。二十年后，牛顿写出了他那本宏伟的《自然哲学的数学原理》，英文全称是《自然哲学的数学原理》；和当时大多数欧洲学者一样，牛顿用拉丁文写作。我要补充一点，“物理学家”一词是十九世纪的；在此之前，理论物理学被称为“自然哲学”。事后看来，两者的区别在于，自然哲学注重逻辑论证，而理论物理学提供可以通过实验检验的解释和预测——这是牛顿率先改变的观点。例如，传统的自然哲学家援引了“逻辑”假设，即某种机械的东西，如巨大的空灵漩涡，将行星带到它们的轨道上。除了它们声称要“解释”的行星运动之外，没有任何实际证据表明这些漩涡存在。相比之下，牛顿根据可观察的事实对行星运动做出了解释，并由此得出了任何两个物体（包括行星和太阳）之间引力的可测试公式：引力与它们质量的乘积成正比，与它们之间的距离成反比。当然，用符号代数形式来表达，这就是著名的平方反比定律$F=\frac{格米米}{r^2}$，其中质量为m和M，假设距离为r，G为比例常数。牛顿也是在《自然哲学的数学原理》中首次发表了他的通用微积分算法，并将其用于他的运动科学中。这些运动定律，加上引力定律，是如此普遍，以至于它们解释了一切从一片树叶的飘落到潮汐的引力，从空中抛出的球的轨迹到行星的运动。这是一项惊人的成就。

Despite Hobbes and his fellow geometers, a decade later it was Wallis’s book that first set young Newton on his path to calculus. Two decades after that, Newton wrote his magnificent Principia, whose full title, in English, is Mathematical Principles of Natural Philosophy; like most European scholars at the time, Newton wrote in Latin. I should add here that the term “physicist” is a nineteenth-century one; before then, theoretical physics was called “natural philosophy.” The difference, in hindsight, is that natural philosophy focussed on logical arguments, while theoretical physics offers explanations and predictions that can be experimentally tested—a change of viewpoint that Newton spearheaded. For example, traditional natural philosophers invoked the “logical” hypothesis that something mechanical, such as gargantuan ethereal whirlpools, carried the planets in their orbits. There was no actual evidence these whirlpools existed, aside from the planetary motion they purported to “explain.” By contrast, Newton had worked out an explanation for planetary motion based on observable facts, from which he derived a testable formula for the gravitational attraction between any two bodies, including a planet and the Sun: it is proportional to the product of their masses and inversely proportional to the distance between them. In symbolic algebraic form this is, of course, the famous inverse square law $F=\frac{GmM}{r^2}$, where the masses are m and M, say, the distance is r, and G is the constant of proportionality. It was also in Principia that Newton first published his general calculus algorithms, which he used in his science of motion. These laws of motion, together with the law of gravity, were so general that they explained everything from the fall of a leaf to the pull of the tides, from the path of a ball thrown in the air to the motion of the planets. It was an astonishing achievement.

爱因斯坦详细解释了这一成就的重要性：“在牛顿之前，”他说，“不存在关于经验世界深层特征的自足的物理因果关系系统。”例如，他补充道，开普勒基于观察的行星运动定律，包括行星轨迹的椭圆形状，确定了行星如何运动，但没有确定为什么运动。这是因为“这些定律关注的是整个运动，而不是系统的运动状态如何引起紧随其后的运动的问题；我们现在应该说，它们是积分定律，而不是微分定律。” ¹¹我们很快就会看到牛顿的微分第二运动定律，稍后我们还会看到，这种对微分定律而非积分定律的偏好是詹姆斯·克拉克·麦克斯韦成功揭开光的秘密的关键。

Einstein explained in detail the importance of this achievement: “Before Newton,” he said, “there existed no self-contained system of physical causality [about] the deeper features of the empirical world.” For instance, he added, Kepler’s observation-based laws of planetary motion, including the elliptical shape of planetary trajectories, established how the planets move, but not why. That’s because “these laws are concerned with the movement as a whole, and not with the question how the state of motion of a system gives rise to that which immediately follows it in time; they are, as we should say now, integral and not differential laws.”¹¹ We’ll see Newton’s differential second law of motion shortly, and later we’ll see that this same preference for differential over integral laws is key to James Clerk Maxwell’s success in uncovering the secret of light.

比这些批评更让牛顿感到苦恼的是罗伯特·胡克 (Robert Hooke) 指控他抄袭。胡克在科学家中名声不佳，因为他喜欢自吹自擂——而且他似乎特别喜欢公开批评牛顿。不过，最近，历史学家对他的作品进行了更仔细的研究，似乎他应该因对行星运动的引力解释提出一些重要的原创性想法而受到赞扬——包括所有行星都有引力，并且引力“可能”与距离成平方反比减弱，因为光的强度就是以这种方式减弱的。尽管牛顿之前已经思考了这个问题很多年，但正是胡克在 1679 年写给他的信激发了他对这个问题的思考——正如他在《数学原理》第一版中承认的那样。胡克提出了轨道运动由两种不同的“运动”组成的想法，一种是切向轨道的运动，另一种是指向运动中心的运动。这是一个直观的想法，即运动是由类似矢量的分量构成的，尽管它并不完全正确：牛顿会表明力是向内的，但运动（速度）是切向的。

Even more distressing for Newton than these criticisms was Robert Hooke’s accusation of plagiarism. Hooke hasn’t had a good rap among scientists, because he tended to big-note himself—and he seemed to take special delight in criticising Newton publicly. Recently, though, historians have taken a closer look at his work, and it seems he deserves credit for some significant original thinking about a gravitational explanation of planetary motion—including the idea that all planets have gravity, and that it weakens with distance “probably” in inverse square proportion, because that is the way light’s intensity weakens. Although Newton had been thinking about the problem for years beforehand, it was Hooke’s letters to him, in 1679, that catalysed his thinking on the subject—as he acknowledged in the first edition of Principia. Hooke suggested the idea that orbital motion was made up of two different “motions,” one tangential to the orbit and one directed to the centre of the motion. It was an intuitive idea of motion being made of vector-like components, although it wasn’t quite correct: Newton would show that the force is directed inward but the motion (velocity) is tangential.

胡克没有将引力扩展到行星以外的物体，他无法通过直接实验或计算证明平方反比定律。相反，他根据对机械模型运动的仔细观察，做出了一些巧妙的类比。12^与牛顿不同，他没有将行星运动推广到所有类型运动的数学定律——因此，他没有像牛顿那样推导出我们现在所说的动能和势能的守恒。（莱布尼茨也开创了能量守恒的概念。）胡克也没有用他的引力概念来估计地球的扁圆形，也没有解释潮汐、春分点进动或牛顿解释的任何其他奥秘。因此，当胡克指责他抄袭时，牛顿非常愤怒：虽然胡克的数学能力比人们长期以来认为的要强，但根本无法与牛顿相提并论。难怪牛顿勃然大怒埃德蒙·哈雷说，如果胡克可以自称是引力理论的发明者，那么“发现、解决并做所有事情的数学家们就只能满足于做枯燥的计算者和苦力”。他说得有道理。有趣的是，在今天的版权法中，受保护的是表达形式而不是思想。尽管最近数学教育者中出现了一个重要举措，强调计算在数学历史和应用中的作用，但许多现代数学家与牛顿保持一致，在他们的工作中看到了美和创造力，而不仅仅是计算和应用^。13

Hooke didn’t extend gravity to material bodies other than planets, and he wasn’t able to prove the inverse-square law by direct experiment or calculation. Rather, he made some ingenious analogies based on careful observation of the motion of mechanical models.¹² And unlike Newton, he didn’t generalise planetary motion to mathematical laws for all types of motion—and consequently he didn’t deduce, as Newton did, the conservation of what we now call kinetic and potential energy. (Leibniz, too, pioneered the notion of conservation of energy.) Nor did Hooke use his idea of gravity to estimate the oblate shape of the Earth or to explain the tides, the precession of the equinoxes, or any of the other mysteries that Newton explained. So, when Hooke accused him of plagiarism, Newton was furious: while Hooke’s mathematical ability was greater than has long been assumed, it was simply no match for Newton’s. No wonder Newton fumed to Edmond Halley that if Hooke could claim to be the inventor of the theory of gravity, then the mathematicians that “find out, settle and do all the business must content themselves with being nothing but dry calculators and drudges.” He had a point. It is interesting that in copyright law today, it is forms of expression rather than ideas that are protected. And although recently there has been an important move among maths educators to emphasise the role of calculation in the history and application of mathematics, many modern mathematicians are in tune with Newton, seeing beauty and creativity in their work, not just computations and applications.¹³

牛顿并不反对计算。他的大部分研究都致力于计算，因为并不总是能够找到方程的精确代数解，更不用说解决更复杂的问题了。所以你必须用数字来处理它，代入数字直到得到一个足够准确的答案——这有点让人想起古老的穷举法。卡尔达诺就是用这种方法得到x = 4 的解，但牛顿开发了不止一种而是几种系统的数值方法。

Not that Newton was averse to computation. Much of his research was devoted to it, for it is not always possible to find exact algebraic solutions of equations, let alone to solve more complex problems. So you have to come at it numerically, plugging in numbers until you get a suitably accurate answer—somewhat reminiscent of the ancient method of exhaustion. Cardano did this to get to his solution x = 4, but Newton developed not one but several systematic numerical approaches.

至于《自然哲学的数学原理》，在它首次出版五十年后，仍有一些人认为牛顿的方法过于数学化——方程式无法像机械模型那样提供因果解释。在法国，牛顿的早期支持者包括埃米莉·杜·夏特莱和她的搭档、煽动性剧作家伏尔泰。18 世纪 30 年代，他们合作出版了一本畅销书，旨在将牛顿的引力和光理论带给更广泛的读者。 （除了相信光是物质粒子之外，牛顿还做了很多细致的实验，包括证明普通阳光是由彩虹中的所有颜色组成的实验。杜·沙特莱进一步阐述了这一观点，她认为不同的颜色与不同的热量有关——这在当时是一个前沿的想法，她通过将床单染成彩虹色并计算每种颜色干燥所需的时间来进行测试：紫色最先干燥，然后是其他颜色，红色干燥时间最长。这是一个漂亮的 DIY 实验，实验表明紫色是最冷的颜色，红色是最暖的颜色，尽管这需要后来的科学家来证实和解释。14 ^）杜·沙特莱还写了一本书，比较了莱布尼茨和牛顿的思想。但她留下的最持久的遗产是她将《自然哲学的数学原理》从拉丁文翻译成法文——这是第一部将《自然哲学的数学原理》翻译成英语以外的日常语言的译本。该译本非常出色，至今仍是权威的法文版。相比之下，最初的英文译本早已被超越。（现代版由 I. Bernard Cohen 和 Ann Whitman 于 1999 年出版。他们发现 du Châtelet 的译本特别有助于澄清一些使用 17 世纪术语的晦涩段落，这些术语后来已被现代化。）

As for Principia, fifty years after its first publication there were still some who thought Newton’s approach was too mathematical—that an equation couldn’t provide a causal explanation the way a mechanical model could. Among Newton’s early champions in France were Émilie du Châtelet and her partner, the provocative playwright Voltaire. In the 1730s, they collaborated on a popular book aimed at bringing Newton’s theories of gravity and light to a wider audience. (Aside from believing that light was a material particle, Newton did many meticulous experiments, including those showing that ordinary sunlight is made up of all the colours of the rainbow. Du Châtelet expanded on this by suggesting that the different colours are associated with different amounts of heat—a cutting-edge idea at the time, which she tested by dyeing a bedsheet in rainbow colours and timing how long it took each colour to dry: violet was first, and then the other colours in order, with red taking the longest. It was a beautiful DIY experiment, suggesting that violet is the coolest colour and red the warmest, although it would take later scientists to confirm and explain it.¹⁴) Du Châtelet also wrote a book comparing the ideas of Leibniz and Newton. But her most enduring legacy is her translation of Principia from Latin to French—the first such translation into an everyday language other than English. It was so good that it is still the definitive French version. By contrast, the original English translation has long since been surpassed. (The modern version is by I. Bernard Cohen and Ann Whitman, published in 1999. They found du Châtelet’s translation particularly helpful in clarifying some obscure passages using seventeenth-century technical terms that have since been modernised.)

杜·夏特莱直到三十岁左右才开始接触数学——作为一个女孩，她没有接受正规教育（就像一个世纪后的索菲·杰曼和玛丽·萨默维尔一样），并在十八岁时嫁给了一位合适的贵族——杜·夏特莱侯爵。她与知识发现和伏尔泰交织在一起的爱情故事非常浪漫，但她作为一名数学家的成长经历也真正鼓舞人心。她在香槟的城堡里建起了房子——以及一个最先进的图书馆和科学实验室——她和伏尔泰吸引了来自四面八方的作家和学者。 （实际上，这是她丈夫的城堡，但他是个正派人，接受了伏尔泰在妻子心中的地位。毕竟，他的婚姻是包办婚姻，不是爱情婚姻，而且他经常外出打仗，就像贵族应该做的那样，还经常和自己的情人调情。此外，伏尔泰还支付了这座古城堡大部分必要的翻修费用。）游客们陶醉于热闹的谈话和伏尔泰专门为他们写的小剧本，以及安静的研究空间。正是这样的一位游客启发了夏特莱翻译《自然哲学的数学原理》，因为她已经成为最新数学物理学的公认权威。

Du Châtelet hadn’t come to mathematics until she was about thirty—as a girl, she’d been denied formal education (just like Sophie Germain and Mary Somerville a century later) and had been married off at eighteen to a suitable aristocrat, the Marquis du Châtelet. The story of her intertwined love affairs with intellectual discovery and with Voltaire is wonderfully romantic, but her development as a mathematician in her own right is truly inspiring. Setting up house—and a state-of-the-art library and scientific laboratory—in her château in Champagne, she and Voltaire attracted writers and scholars from far and wide. (Actually, it was her husband’s château, but he was a decent man who had come to accept Voltaire’s place in his wife’s heart. After all, his had been an arranged marriage, not a love match, and he was often away fighting wars, as aristocrats were expected to do, and dallying with his own lovers. Besides, Voltaire paid for much of the necessary renovation of the old castle.) Visitors reveled in the lively conversation and the little plays written especially for them by Voltaire, as well as the peaceful space for research. It was one such visitor who inspired du Châtelet to translate Principia, for she had become a recognised authority on the latest mathematical physics.

尽管如此，在女性没有知识分子地位的时代，对于一个主要靠自学的女性来说，这仍然是一项令人难以置信的事业。起初，她不得不自学数学——尽管后来她聘请了两位法国最优秀的数学家皮埃尔-路易·莫罗·莫佩尔蒂和亚历克西斯-克劳德·克莱罗作为导师。伏尔泰也参加了一些课程，但他很快就屈服于她在数学上的优越性，开玩笑地称她为“夏特莱的牛顿夫人”。

Still, it was an incredible undertaking for a mostly self-taught woman at a time when women had no intellectual standing. Initially she had had to teach herself maths—although later she employed as tutors two of France’s best mathematicians, Pierre-Louis Moreau Maupertuis and Alexis-Claude Clairaut. Voltaire had joined some of the lessons, but he soon deferred to her mathematical superiority, playfully dubbing her “Madame Newton du Châtelet.”

除了翻译牛顿的 500 多页原文外，夏特莱夫人还提供了一份 180 页的附录，其中包括 110 页的牛顿以来引力理论发展的概述（包括她的朋友莫佩尔蒂和克莱罗的研究），以及 70 页的附加数学知识，旨在阐明牛顿的工作原理。众所周知，牛顿选择使用创新的几何构造而不是符号微积分计算来证明他的大部分定理（即使在使用微积分时也是如此）。15^也许霍布斯的“符号疤痕”仍然引起共鸣。问题是，牛顿的证明如此巧妙和独特，以至于它们不能轻易地被推广——这意味着他的运动思想不能轻易地扩展到新的应用和见解。一代又一代的学者都对他感到失望：他发明了微积分，那么为什么不更透明地使用它呢？显然，他认为几何是当时最严格、最直观的数学形式。

In addition to translating Newton’s 500-odd pages, du Châtelet provided a 180-page appendix. This included a 110-page outline of developments in gravitational theory since Newton—including those by her friends Maupertuis and Clairaut—and 70 pages of additional mathematics, designed to clarify Newton’s working. He had famously chosen to present the proofs of most of his theorems using innovative geometrical constructions rather than symbolic calculus calculations—even when he was using calculus.¹⁵ Perhaps Hobbes’s “scab of symbols” still resonated. Trouble was, Newton’s proofs were so clever and idiosyncratic that they couldn’t readily be generalised—which meant that his ideas of motion couldn’t readily be extended to new applications and insights. Generations of scholars have been frustrated with him: He invented calculus, so why didn’t he use it more transparently? Evidently, he thought geometry was the most rigorous and intuitive form of mathematics at that time.

例如，将速度等导数以几何学的形式想象成三角形斜边的斜率，其边代表距离和时间的无穷小变化，这当然非常有帮助。图 2.4显示了这个想法，以及莱布尼茨的代数符号——我们熟悉的$\frac{d是}{d十}$—反映了这种几何的、直观的解释。

For example, it is certainly very helpful to imagine a derivative, such as speed, geometrically—as the gradient of the hypotenuse of a triangle whose sides represent the infinitesimal changes in distance and time. Figure 2.4 shows the idea, and Leibnizian algebraic symbolism—the familiar $\frac{dy}{dx}$—reflects this geometrical, intuitive interpretation.

然而，在现代导数概念中，莱布尼茨符号有点误导，因为$\frac{d是}{d十}$根本不是比率或分数，不是dy除以dx的数。相反，$\frac{d}{d十}$是作用于函数 y ( x ) 的算符——用现代语言来说；它只是意味着符号$\frac{d}{d十}$指导你通过求导数来“操作” y ( x )。然而，莱布尼茨的导数符号可以像处理普通分数一样进行操作。一个经典的例子是“链式法则”，你可以写出如下方程$\frac{d是}{d十}\frac{d十}{d吨}=\frac{d是}{d吨}$，它确实看起来就像您只是取消了dx一样。

In the modern conception of a derivative, however, Leibnizian notation is a little misleading, because $\frac{dy}{dx}$ is not a ratio or fraction at all, in the sense of a number dy divided by a number dx. Rather, $\frac{d}{dx}$ is an operator acting on a function y(x)—to use modern language; it just means that the symbol $\frac{d}{dx}$ directs you to “operate” on y(x) by taking its derivative. Nevertheless, the Leibnizian symbols for derivatives can be manipulated as if they really are ordinary fractions. A classic case is the “chain rule,” where you can write equations such as $\frac{dy}{dx}\frac{dx}{dt}=\frac{dy}{dt}$, which does look for all the world as if you simply canceled the dx’s.

牛顿对严谨性更加谨慎，他没有这样的透明的象征主义。例如，用莱布尼茨符号，你可以写$五d吨=\frac{d十}{d吨}d吨=d十$，这很容易表明，以速度v运动的物体在dt瞬间行进了微小的距离dx。相比之下，在牛顿的符号中，这将是vo = ẋo，因此从符号上看，物理解释并不那么明显。无论如何，他似乎认为，如果他坚持字面上的几何构造，《数学原理》会更容易理解：了解引力理论会引起争议，他不想在微分消失和芝诺悖论仍未解决的情况下过多地研究新微积分。当他需要微积分时，他倾向于将其写成图下的面积和几何比率，就像这样在图 2.4中，尽管他有时确实使用代数来解释他的算法和图表（正如我在前面的尾注中所示，即第 15 号）。

Newton was more cautious about rigour, and he didn’t have such a transparent symbolism. For instance, with Leibnizian symbols you can write $vdt=\frac{dx}{dt}dt=dx$, which readily shows that an object moving at speed v travels a tiny distance dx in the instant dt. By contrast, in Newton’s notation this would be vo = ẋo, so a physical interpretation is not so obvious from the symbolism. At any rate, he seemed to think that Principia would be more accessible if he stuck to literal geometric constructions: knowing the theory of gravity would be controversial, he didn’t want to make too much of the new calculus with its disappearing differentials and with Zeno’s paradoxes still unresolved. When he needed calculus, he tended to write it in terms of areas under graphs and geometric ratios like that in figure 2.4, although he did sometimes use algebra to explain his algorithms and diagrams (as I illustrated in the previous endnote, i.e. no. 15).

半个世纪后，最初得益于瑞士数学家约翰·伯努利（或让·伯努利），微积分符号系统至少在欧洲大陆变得更加被接受。在莱布尼茨与牛顿之间臭名昭著的优先权之争中，伯努利是莱布尼茨最热心的辩护人，后来他与杜·夏特莱的圈子保持着联系。在《数学原理》（法语版《数学原理》）的附录中，杜·夏特莱用代数微积分重新阐述了牛顿的几十个证明。而且，英国人无疑认为这是一种不受欢迎的讽刺，她用的是莱布尼茨的符号系统，而不是牛顿的^。16

Half a century later, thanks initially to the formidable Swiss mathematician Johann (or Jean) Bernoulli, calculus symbolism had become more acceptable—on the continent, at least. Bernoulli had been Leibniz’s most avid defender during the infamous priority dispute with Newton, and later he’d corresponded with du Châtelet’s circle. In the appendix to Principes Mathématiques (her French Principia), du Châtelet reworked dozens of Newton’s proofs in terms of algebraic calculus. And, in what the British no doubt saw as an unwelcome irony, she did it using Leibniz’s symbolism, not Newton’s.¹⁶

然而，通过将牛顿的几何微积分转化为莱布尼茨符号，杜·夏特莱和伯努利及其学生帮助将牛顿的愿景完全展现出来。正是由于他们以及牛顿，微积分才在物理学和技术领域中发挥了如此重要的作用。

Yet, by translating Newton’s geometrical calculus into Leibnizian symbols, du Châtelet and Bernoulli and his students helped bring Newton’s vision into full view. It is thanks to them, as well as to Newton, that calculus has proved so important in physics and technology.

• • •

• • •

然而，在本书的其余部分，我们将探讨矢量和张量微积分，以及矢量和张量本身的思想和应用。我刚才谈到了速度作为导数，但正如我在序言中提到的，“速度”是一个更精确的量：它给出了运动的方向和速度，因此它包含有关两个不同事物的信息。这就是汉密尔顿所说的“矢量”。速度本身，就像任何“标量”或数字一样，只代表一个属性。

It is vector and tensor calculus, however, that we’ll explore in the rest of this book, along with the ideas and applications of vectors and tensors themselves. I spoke just now about speed as a derivative, but as I mentioned in the prologue, “velocity” is a more precise quantity: it gives the direction of the motion as well as the speed, so it contains information about two different things. It is what Hamilton called a “vector.” Speed alone, like any “scalar” or number, represents just one attribute.

这样说来，向量的概念似乎非常简单。但正如前面几节所暗示的，它对数学家、物理学家和工程师的用处也取决于它的符号表示。那么，如何表示和计算向量呢？这是我们现在要开始探讨的一个主题，以及各种有趣问题的答案。例如，如果向量真的如此简单，那么汉密尔顿在布鲁姆桥的顿悟有什么重要意义呢？

When you put it like this, the concept of a vector seems very simple. As the preceding sections have suggested, though, its usefulness to mathematicians, physicists, and engineers will also hinge on its symbolic representation. So, how to represent, and calculate with, a vector? That’s a topic we’ll now begin to explore—along with answers to various intriguing questions. For instance, if vectors really are so simple, what was so important about Hamilton’s epiphany at Broome Bridge?

现在是时候更深入地探究这个故事了。

It is time to dive more deeply into the story.

由于数学家并不以涂鸦闻名，威廉·罗文·汉密尔顿在布鲁姆桥上的涂鸦引起了很多关注。但是，这种导致这种狂热的轻微破坏行为的灵光一现的灵感，很少没有经过长期深刻的创造性思考。就向量而言，汉密尔顿花了数百年的时间，经过许多数学思想家的努力，才将所有的想法都整合在一起，最终得出了他自己深刻的结论。所以，让我从重温令人敬畏的牛顿开始这个故事的这一章。

Since mathematicians are not known for their graffiti, William Rowan Hamilton’s scratchings on Broome Bridge have attracted a lot of attention. But the kind of lightning-bolt inspiration that led to this ecstatic act of minor vandalism rarely comes without a long history of deep, creative thinking. In the case of vectors, it took hundreds of years, and many mathematical thinkers, for all the ideas to fall into place so that Hamilton could score his own deep conclusions in stone. So let me begin this chapter of the story by revisiting the redoubtable Newton.

这些天他并不总是表现得很好，但最近有人问我是否想请他共进晚餐。我已经提到过我钦佩他的天才，但我也喜欢他的人性化小细节——例如，他会把书折角，并且全神贯注于工作而忘了吃饭。还有，那棵传奇的苹果树仍然生长在他家的花园里，并在世界各地都有后代——包括我自己的大学。但更重要的是，我把这次与牛顿坐在一起的邀请当作一个重新研究《自然哲学的数学原理》的机会，我第一次不仅被它的范围和它对微积分的开创性应用所震撼，而且被它对“矢量”数量的基本性质的清晰阐述所震撼。

He doesn’t always come off well these days, but recently I was asked if I’d like to invite him to dinner. I’ve mentioned already my admiration of his genius, but I also like the little human touches—dog-earing his books, for example, and becoming so engrossed in his work that he forgot to eat. Then there’s the fact that the legendary apple tree is still growing in the garden of his family home and has descendants around the world— including at my own university. More seriously, though, I took this invitation to sit with Newton as a chance to dip back into Principia, and for the first time I was struck not just by its scope and its pioneering applications of calculus, but by its clear exposition of the fundamental nature of “vectorial” quantities.

我以前从未关注过这个问题——我只是想当然地认为向量，所以我没有意识到牛顿的处理有多么重要。我没有意识到，在为基于证据、可测试的自然物理理论制定蓝图时，他看到了定义基本物理量的必要性，例如力、速度和动量，以便它们既有方向又有大小。正如我们所见，今天的高中数学学生学会用箭头来表达这种双重性质。但矢量本身的出现却花了很长时间，甚至连牛顿都没有完全理解这个概念。

I’d never paid this any attention before—I’d simply taken vectors for granted, so I hadn’t realised how significant Newton’s treatment was. I hadn’t realised how telling it is that, in creating the blueprint for evidencebased, testable physical theories of nature, he saw the need to define fundamental physical quantities, such as force, velocity, and momentum, so that they have both direction and magnitude. Today’s high school maths students learn to express this dual nature as an arrow, as we’ve seen. But the vector itself was a surprisingly long time in coming, and not even Newton got the full concept.

他在这方面的关键见解是从力对物体的影响的角度对力进行定义。在他著名的第二运动定律中，他说，有效力与由此产生的“运动改变”成正比，而这种“改变”与力的方向相同。使用文字形式的方程，牛顿已经对“运动量”（他在第二定律中称之为“运动”）给出了仔细的定义，即“物质的量”（质量）乘以速度——因此在符号中运动量为mv，我们称之为“动量”。因此，用现代语言来说，牛顿的“运动改变”是物体动量的变化率，用矢量微积分来说，第二定律为：

His key insight in this context was his definition of force in terms of its effect on material bodies. The effective force, he said in his famous second law of motion, is proportional to the resulting “alteration of motion,” and this “alteration” is in the same direction as the force. Using word-form equations, Newton had already given a careful definition of “quantity of motion” (which he just called “motion” in the second law) as the “quantity of matter” (mass) times the velocity—so in symbols the quantity of motion is mv, which we call “momentum.” In modern language, then, Newton’s “alteration of motion” is the rate of change of the body’s momentum, and in terms of vector calculus, the second law is:

$F=\frac{d\left(米五\right)}{d吨}$。

$F=\frac{d\left(mv\right)}{dt}$.

如果物体的质量在运动过程中没有发生可测量的变化（除非相对于测量者来说物体移动得非常快），那么这就相当于$F=米\frac{d五}{d吨}$，或者说质量乘以加速度，F = m a。我用粗体字写了F、a和v，以表明它们是矢量，但这是事后才知道的。虽然很明显，牛顿的力和动量既有大小又有方向，但它们还不是完整的数学矢量，因为没有如何将它们相加或相乘的一般概念。你可能会认为，担心相加和相乘是一种神秘的区别，因为矢量最明显的特征是它能够同时表示大小和方向。然而，正如我们将看到的，矢量相乘开辟了一系列全新的物理应用。正是矢量的代数，被视为数学量，而不是简单的物理类似物，才使得矢量不仅适用于理论物理学，还适用于数值建模、数据分析、工程、人工智能和机器人等领域。

If the body’s mass doesn’t change measurably during the motion— which it doesn’t unless it is moving extremely fast relative to the measurer—then this is the same as $F=m\frac{dv}{dt}$, or mass times acceleration, F = ma. I’ve written the F, a, and v in bold type to indicate that they are vectors, but this is hindsight. Although it was clear that Newton’s force and momentum had both magnitude and direction, they were not yet full mathematical vectors, for there was no general concept of how to add or multiply them. You might think that worrying over adding and multiplying is an arcane distinction, given that the most obvious characteristic of a vector is its ability to represent both magnitude and direction at the same time. As we’ll see, though, multiplying vectors opens up a whole new range of physical applications. And it is the algebra of vectors, treated as mathematical quantities rather than simply as physical analogues, that makes vectors applicable not only in theoretical physics, but also in fields such as numerical modeling, data analysis, engineering, AI, and robotics.

与此同时，在牛顿时代，有简单加法这一实用概念，即两个矢量通过图 3.1所示的“平行四边形规则”以几何方式相加。实际上，这里的“加法”一词在矢量正式发展之前可能已经过时了——“合成”才是更准确的说法。在牛顿运动定律的前两个推论中，牛顿对这一规则做出了清晰的解释，并明确展示了它如何使对角线方向的力由另外两个“分”力合成（反之，也可以“分解”为其分力）。当然，平行四边形规则也适用于速度，牛顿在其行星和日常运动理论中对其进行了精彩的运用。

Meantime, in Newton’s day there was a practical notion of simple addition, where two vectorial quantities were added geometrically via the “parallelogram rule” shown in figure 3.1. Actually, the term “addition” here is probably anachronistic before the formal development of vectors— “composition” is a more accurate term. In the first two corollaries to his laws of motion, Newton gave a clear explanation of this rule, and showed explicitly how it enables a force in the diagonal direction to be composed from two other “component” forces (and conversely, it can be “decomposed” into its components). The parallelogram rule applies to velocities, too, of course, and Newton went on to make spectacular use of it in his theories of planetary and everyday motion.

“运动”这个概念看似简单，但正如芝诺所言，想出一个有用的定义并不容易，尤其是一个能让你提前预测物体在各种力的作用下如何运动的定义。牛顿的对手莱布尼茨在 1695 年的《动力学论文》中花费了大量篇幅试图弄清楚这一点，但他没有提到《数学原理》 。牛顿后来以牙还牙，从《数学原理》第三版中删除了对莱布尼茨微积分的认可——不幸的微积分优先权之争如火如荼地展开。莱布尼茨在《动力学论文》中对牛顿的忽视不仅仅是但这未免有些粗鲁：它表明了理解新思想是多么困难。莱布尼茨也许是他那一代最伟大的哲学家，也是一位创新的科学思想家，但就连他似乎也没有意识到，当牛顿用由此产生的动量变化来定义力时，他在运动分析方面取得了巨大的进步。（当然，后来的物理学家确实承认了这一成就，并最终将力的 SI 单位¹命名为“牛顿”。）

It seems such a simple concept, “motion,” but, as Zeno showed, it’s not so easy to come up with a useful definition—especially one that allows you to predict in advance how a body will move under various forces. Newton’s rival Leibniz spent many wordy pages trying to figure it out, in his 1695 Essay on Dynamics—but he didn’t mention Principia. Newton later returned the favour by removing his acknowledgment of Leibniz’s calculus from Principia’s third edition—the unfortunate calculus priority dispute was in full swing. Leibniz’s neglect of Newton in his Essay was more than churlishness, though: it shows how difficult it can be to understand new ideas. Leibniz was perhaps the greatest philosopher of his generation, and an innovative scientific thinker, yet even he didn’t seem to realise that Newton had made a great step forward in the analysis of motion when he defined force in terms of the resulting change in momentum. (Later physicists did recognise this achievement, of course, and ultimately named the SI unit¹ of force the “newton.”)

然而，有趣的是，即使是牛顿也没有提出向量代数的概念，更不用说向量微积分了。这凸显了提出新的数学概念有多么困难——也表明向量看似简单，其实是骗人的。

It is intriguing to realise, though, that not even Newton developed the idea of vector algebra, let alone vector calculus. It highlights just how hard it is to come up with new mathematical concepts—and it suggests that the seeming simplicity of vectors is deceptive.

• • •

• • •

事实上，当牛顿写作《自然哲学的数学原理》时，他已经经过了漫长的道路，集结了世界上最优秀的思想家，才得出平行四边形定律。像牛顿和莱布尼茨一样，这些思想家也一直在努力理解事物如何运动以及力如何作用。想象一下如何从头开始定义“力”，你就能体会到他们所面临的困难。牛津英语词典给出的日常定义是“力量；施加的力量或动力；巨大的努力”。这给了你大概的概念，但并不能帮助你发射通讯卫星或设计风力涡轮机。对于这类应用，你需要一个定量的矢量微积分定义。

In fact, when Newton was writing Principia, it had already been a long road, peopled with some of the best thinkers in the world, just to arrive at the parallelogram rule. Like Newton and Leibniz, these thinkers had been trying to work out how to understand how things move and how forces act. You can get a sense of the difficulty they faced by imagining how you would go about defining “force” from scratch. The everyday definition given in the Oxford English Dictionary is “power; exerted strength or impetus; intense effort.” That gives you the general idea, but it’s not going to help you launch a communications satellite or design a wind turbine. You need a quantitative, vector calculus definition for that kind of application.

当然，很明显，如果你用两倍的力推一个物体，它就会移动两倍远——这是古代哲学家很久以前就采取的量化步骤。不过，在这种情况下，两次推力的方向与产生的运动方向相同，因此方向和强度不应该成为力本身定义的一部分，这一点并不明显。但古人确实瞥见了从不同方向的两个不同“分量”组成轨迹的想法——例如在托勒密的行星运动机械模型中，行星旋转就像它们由齿轮驱动一样，齿轮也沿着轨道缓慢滚动。

Of course, it’s obvious that if you push twice as hard on an object it will move twice as far—a quantifying step that ancient philosophers had taken long ago. In this case, though, both pushes are in the same direction as the resulting motion, so it isn’t apparent that direction as well as strength should be part of the definition of force itself. But the ancients did glimpse the idea of composing a trajectory from two different “components” in different directions—such as in Ptolemy’s mechanical model of planetary motion, where the planets whirled about as if they were powered by gears that also rolled slowly along their orbits.

通过平行四边形规则将不同的运动结合起来的第一个暗示似乎是在《机械问题》中，它显然是由一位他是公元前四世纪末亚里士多德学派的成员，比托勒密早了近五百年。这是一份失传已久的手稿，直到文艺复兴时期，在中世纪伟大的阿拉伯翻译运动之后，才在欧洲重新发现——其中对力学的处理启发了塔尔塔利亚和伽利略等早期现代物理学家。他们试图找出如何从分运动中组合出移动物体的路径，例如当网球抛向空中时，斜向上运动和重力向下作用的组合^。2

The first hint of combining different motions via the parallelogram rule seems to be in Questiones Mechanicae, which was apparently written by a member of Aristotle’s school at the end of the fourth century BCE, nearly five hundred years before Ptolemy. It was one of those long-lost manuscripts that were only rediscovered in Europe during the Renaissance, in the wake of the great medieval Arabic translation movement—and its treatment of mechanics proved inspirational to early modern physicists such as Tartaglia and Galileo. They were trying to work out how to compose the paths of moving objects from component motions, such as the combination of an oblique upward motion and the downward effect of gravity when a tennis ball is lobbed into the air.²

实际上，塔尔塔利亚和伽利略对网球的兴趣远不及箭和炮弹；当时的宗教和帝国战争似乎永无休止。枪支和大炮变得越来越复杂——例如，新的铸铁技术意味着炮弹现在更轻但威力更大——而且越来越明显的是，提前知道导弹的轨迹将使这些新武器能够更有效地部署。炮手手册也出版了，记录了观察到的各种射击的射程和角度，数学家开始使用这些数据试图用数学方法描述射弹的路径。

Actually, Tartaglia and Galileo were interested not so much in tennis balls as in arrows and cannonballs; such were the times, with their seemingly endless religious and imperial wars. Guns and cannon were becoming more sophisticated—for example, new cast-iron techniques meant cannonballs were now lighter but more powerful—and it was increasingly clear that knowing a missile’s trajectory in advance would enable these new weapons to be deployed more efficiently. Gunners’ manuals were being published, too, setting down records of the observed ranges and angles of various shots, and mathematicians began to use these data to try to describe the path of a projectile mathematically.

首先，在 1530 年代早期——在与卡尔达诺就三次方程展开激烈争论的几年前——塔塔利亚设法算出，如果想让炮弹飞得尽可能远，大炮应与地面呈 45° 角。如果你只是将炮弹直直地指向远处的目标，重力可能会让你无法击中目标。如果你将炮弹指向高处并希望获得最佳效果——这早已是惯常做法——那么你通常会错过目标并摧毁其他东西。但塔塔利亚遭受了良心危机，因为他知道他在研究战争武器时是在拿人们的生命开玩笑。因此，他烧毁了自己的论文，担心上帝会因他从事这种“恶毒而残酷”的工作而发怒^。3

As a first step, in the early 1530s—a few years before his acrimonious quarrel with Cardano over cubics—Tartaglia managed to work out that if you want a cannonball to go as far as possible, the cannon should be pointed at an angle of 45° to the ground. If you just pointed it straight ahead at a distant target, gravity would likely catch you short. And if you pointed it high and hoped for the best—which had long been the usual practice— then you’d often miss your target and destroy something else instead. But Tartaglia suffered a crisis of conscience, for he knew he was dallying with people’s lives while he worked on weapons of war. So he burned his papers, fearing God’s wrath for engaging in such “vituperative and cruel” work.³

然而，那是一个可怕的时代。塔尔塔利亚在意大利北部长大，小时候被一名法国士兵的军刀刺伤，伤势严重。当时他十二三岁，和母亲、妹妹躲在一座教堂里，当时法国人正在攻打他们的城镇，但那把锋利的剑尼科洛的嘴和下巴被一枪打中。这枪的伤痕留在了他的名字塔尔塔利亚身上——他的真实姓氏似乎是丰塔纳，但在上颚受伤后，他得到了“塔尔塔利亚”的绰号，即口吃者。尽管如此，尽管他缺乏教育，他还是成为了十六世纪初最优秀的数学家之一。至于他在弹道学方面的工作，他很快决定，作为一个虔诚的基督徒，他最好和他的赞助人谈论它，因为伊斯兰奥斯曼帝国正在进一步向基督教欧洲扩张。它已经扩展到希腊和土耳其的大部分地区，最北延伸到贝尔格莱德和索非亚，但到了 1530 年代，苏莱曼皇帝的军队入侵了匈牙利，并与奥地利发生了冲突。奥地利哈布斯堡王朝也在扩大自己的影响力，法国和奥斯曼帝国结成联盟对抗他们——就这样，战争似乎无处不在。

These were fearful times, however. As a child growing up in northern Italy, Tartaglia had suffered a terrible wound from a French soldier’s saber. He was twelve or thirteen and had been hiding with his mother and sister in a church as the French stormed their town, but the slashing sword caught young Niccolò across the mouth and jaw. The wound’s legacy lives on in his name, Tartaglia—his actual surname seems to have been Fontana, but after the injury to his palate he acquired the nickname “Tartaglia,” the Stammerer. Despite all this, and despite his lack of education, he became one of the best mathematicians of the early sixteenth century. As for his work on ballistics, he soon decided that as a good Christian he’d better talk about it to his patron, for the Islamic Ottoman Empire was expanding ever further into Christian Europe. It already extended through much of Greece and Turkey, and as far north as Belgrade and Sofia, but by the 1530s Emperor Süleyman’s troops had invaded Hungary and were skirmishing with Austria. The Austrian Habsburgs were extending their own influence, too, and the French and the Ottomans formed an alliance against them— and so it went: war seemed to be everywhere.

于是塔尔塔利亚继续研究这个问题。他按照建议将大炮倾斜 45° 度，设计了一种象限来测量这种角度。他还试图构建一个理论弹道。直观上看，如果忽略空气阻力（对于炮弹来说这是一个足够现实的假设，但对于羽毛来说则不然），这样的路径仅由两个不同的分量构成：一个在发射方向上，一个在重力方向上。但塔尔塔利亚的尝试没有成功。首先，他的分量不是明确定义的力或速度，而只是直观的“运动”——毕竟，他写作的时间比牛顿严谨、开创性的运动定律早 150 年。更重要的是，塔尔塔利亚从未以矢量方式添加（或“构建”）这些“运动”。在对重新发现的《机械问题》的评论中，他只是忽略了平行四边形规则——像他的大多数同时代人一样，他不理解它的重要性^。4

So Tartaglia kept working on the subject. He followed up his advice about angling cannon at 45° by designing a kind of quadrant for measuring such angles. He also tried to compose a theoretical ballistic trajectory. It was intuitively evident that if you ignored air resistance—a realistic enough assumption for a cannonball although not for a feather—such a path was built from just two different components: one in the direction the shot was fired, and one in the direction of gravity. But Tartaglia’s attempt wasn’t successful. First, his components were not clearly defined forces or velocities but just intuitive “motions”—after all, he was writing 150 years before Newton’s careful, groundbreaking laws of motion. What’s more, Tartaglia never added (or “composed”) these “motions” vectorially. In his commentary on the rediscovered Questiones Mechanicae, he’d simply ignored the parallelogram rule—like most his contemporaries, he didn’t understand its significance.⁴

其次，他没有独立分量的概念，即每个分量都像其他分量不存在一样起作用。这是平行四边形规则的关键，如图 3.1所示。举个例子，如果你在这个分析中去掉重力，只让其中一个分量起作用——大炮发射方向的分量——你应该推断出炮弹会沿着初始方向以相同的速度继续移动。这是牛顿第一运动定律，尽管其他人，尤其是伽利略和哈里奥，在他之前就已经直觉地发现了它。

Second, he didn’t have the idea of independent components, each one acting as if the other weren’t there. This is key to the parallelogram rule, as you can see in figure 3.1. An example of what this means is that if you took away gravity in this analysis so that only one of the components acted—the one in the direction the cannon was fired—you should deduce that the ball would keep moving at the same speed along that initial direction. This is Newton’s first law of motion, although others, notably Galileo and Harriot, had intuited it before him.

然而，就像亚里士多德之后的几乎所有人一样，塔尔塔利亚认为物体不可能有两个独立的运动分量同时作用。所以他没有想到通过平行四边形规则将这两个分量结合起来，从而得到最终的观测运动。如图 3.2所示，他的轨迹由一个初始的倾斜分量和一个向下的分量组成。他并不认为两者从一开始就是一起作用的，所以炮弹的路径实际上是弯曲的，而不是直线的，因为重力已经将它拉下了。相反，在轨迹的中间，他有一个弯曲的部分，代表当重力“突然”克服了最初的向外力时炮弹的路线改变，然后炮弹垂直下落。

Tartaglia, however—like almost everyone since Aristotle—believed it was not possible for a body to have two independent components of motion acting at the same time. So he didn’t have the idea of combining the two components, via the parallelogram rule, to give the resulting observed motion. As you can see in figure 3.2, he composed his trajectory from an initial oblique component followed by a downward one. He didn’t have the idea that both were acting together from the outset, so that the ball’s path actually starts out curved, not straight, because gravity is already pulling it down. Instead, in the middle of his trajectory, he has a curved section representing the cannonball’s changing course when gravity “suddenly” overcomes the initial outward force, and then the ball drops straight down.

虽然塔塔利亚的著作是首次对弹道学进行重大研究，但他分析中的缺陷表明，没有人真正知道引力是如何运作的，更不用说理解矢量了。半个世纪后，伽利略和哈里奥特出现了。战争仍在肆虐，但现在新教徒和天主教徒之间的紧张关系带来了新的动乱。然而，人们对“战争艺术”也产生了一种智力和审美上的迷恋。战争游戏是哈里奥特的第二位赞助人诺森伯兰伯爵等人的一种娱乐消遣；他曾在军队服役一段时间，对军事战略如此着迷，以至于他花了无数的时间玩一种精心设计的棋盘游戏，其中有 140 名黄铜士兵，每个士兵都拿着一根铁丝长矛，还有 320 名铅士兵拿着小火枪。战争也是艺术的主题：欧洲大陆的文艺复兴时期艺术家，如乔治奥·瓦萨里和阿尔布雷希特·阿尔特多弗一直在描绘鸟瞰式的战争场景，而现在，在英国，威廉·莎士比亚和克里斯托弗·马洛通过强有力的语言和巧妙的舞台表演，在《亨利五世》和《帖木儿》等戏剧中唤起了人们对大规模战争的回忆。《帖木儿》戏剧化地描述了十四世纪鞑靼战士帖木儿的可怕征服。它或许让人想起了苏莱曼最近的功绩，尽管马洛是在更直接的危险——西班牙无敌舰队的阴影下创作的，西班牙无敌舰队因各种长期酝酿的宗教、地缘政治和经济原因于 1588 年袭击了英国。有证据表明马洛认识哈里奥特——有人推测他们讨论过“战争艺术”，马洛是诗人和剧作家，哈里奥特是研究弹道的数学家^。5

While Tartaglia’s was the first significant foray into the study of ballistics, the flaw in his analysis suggests that no one really knew how gravity operated, let alone understood vectors. Enter Galileo, and Harriot, half a century later. War was still raging, but now the tension between Protestants and Catholics was bringing new unrest. Yet there was an intellectual and aesthetic fascination with “the art of war,” too. War games were an entertaining pastime for people such as Harriot’s second patron, the Earl of Northumberland; he’d served briefly in the military and was so fascinated by military strategy that he spent countless hours playing an elaborate board game with 140 brass soldiers, each with a wire pike, and 320 lead soldiers carrying tiny muskets. War was a subject of art, too: continental Renaissance artists such as Georgio Vasari and Albrecht Altdorfer had been painting bird’s-eye battle scenes, and now, in England, William Shakespeare and Christopher Marlowe were evoking huge battles through powerful language and clever staging, in plays such as Henry V and Tamburlaine. Tamburlaine dramatises the terrifying conquests of the fourteenthcentury Tartar warrior Tamerlane. It was a reminder, perhaps, of the more recent exploits of Süleyman, although Marlowe wrote it in the shadow of a more immediate danger, the Spanish Armada, which for various longsimmering religious, geopolitical, and economic reasons attacked England in 1588. There is some evidence that Marlowe knew Harriot—and it has been conjectured that they discussed the “art of war,” Marlowe as a poet and playwright, and Harriot as a mathematician working on ballistics.⁵

没有证据表明伽利略和哈里奥特彼此认识，但他们对当时最前沿的科学有着共同的兴趣——不仅仅是在弹道学方面，而且在所有受重力影响的运动方面。结果，他们独立发现了自由落体的一般规律——在日常情况下，如果消除空气阻力，所有物体，无论其重量和大小，都会以相同的速度下落。但对于塔尔塔利亚来说，炮弹比网球下落得更快，所以它们的抛射轨迹形状不同。但哈里奥特和伽利略表明，在真空中，所有抛射物​​的轨迹形状相同：抛物线。如果你倾斜一根喷出水流的软管，你可以很容易地看到这个形状，因为每一滴水滴都描绘出抛射物的形状。证明它是一条抛物线，有形式为y = −ax2 + bx + c的^独特方程，则是另一回事。但是，如果你在高中或大学学习过数学或物理（或者如果你提前看过图 8.1），你就会知道，使用牛顿定律、向量和微积分规则，只需十几行代码就可以完成。如果没有这些技巧，伽利略和哈里奥特就花了很多页纸和许多艰苦的月份来寻找适合他们数据的正确方程式。

There’s no evidence that Galileo and Harriot knew each other, but they shared the cutting-edge scientific interests of their time—and not just in ballistics, but in all motion that is affected by gravity. As a result, they independently codiscovered the general law of free fall—that in everyday situations, if you take away air resistance, all bodies, no matter their weight and size, fall at the same rate. For Tartaglia, though, a cannonball fell faster than a tennis ball, so they’d each have differently shaped projectile trajectories. But Harriot and Galileo showed that in a vacuum, all projectiles trace the same shaped path: a parabola. You can see this shape easily if you tilt a hose with a jet of water coming out, because each droplet, one after the other, traces out the shape of a projectile. Proving it’s a parabola, with a distinctive equation of the form y = −ax² + bx + c, is another matter. But if you’ve studied maths or physics at high school or college level (or if you peek ahead at fig. 8.1), you’ll know that it can be done in a dozen lines using Newton’s laws, vectors, and the rules of calculus. Without these techniques, it took Galileo and Harriot many pages and many arduous months as they searched for the right equation to fit their data.

相比之下，你可以在图 3.2中看到，塔尔塔利亚估计的轨迹绝对不是抛物线！伽利略和哈里奥特之所以成功，是因为他们意识到分力运动确实独立起作用——这也是他们如何直观地理解牛顿第一定律的。6^伽利略对抛射体的研究影响了牛顿，牛顿在《自然哲学的数学原理》中引用了伽利略的研究。但伽利略从未真正理解完全平行四边形规则。他用它计算合力的大小，但没有计算方向——大多数其他当代先驱也是如此。勒内·笛卡尔确实用这个规则来计算大小和方向，但他认为它们是独立的量，而不是单个量（如速度或力）的独立方面。所以，你可以看到为什么矢量比我们后见之明和习惯所能意识到的要多得多！⁷

By contrast, you can see in figure 3.2 that Tartaglia’s estimated trajectory was definitely not a parabola! Galileo and Harriot succeeded because they realised that component motions do act independently—which is also how they came to intuit Newton’s first law.⁶ Galileo’s work on projectiles influenced Newton, who cited it in Principia. But Galileo never quite got to the full parallelogram rule. He used it to calculate the magnitudes of his resultant forces, but not the directions—and the same was true for most other contemporary pioneers. René Descartes did use the rule to calculate magnitudes and directions, but he thought of them as separate quantities, not as separate aspects of a single quantity, such as velocity or force. So, you can see why there is more to vectors than we, with our hindsight and habit, might realise!⁷

不过，哈里奥特在 1619 年对碰撞力学的分析中接近了这个想法。除了战争和战争游戏，台球游戏也风靡一时——至少对哈里奥特的赞助人诺森伯兰这样的富裕贵族来说是这样。不幸的是，伯爵很快就被关进了伦敦塔，成为詹姆斯一世国王的囚徒——他和哈里奥特在 1605 年火药阴谋被挫败时被捕，因为他们在前一天不知情的情况下与其中一名阴谋者共进晚餐。（阴谋者是心怀不满的天主教徒，对新教国王及其议会的统治感到不满，其中一名恰好是诺森伯兰的表弟。）哈里奥特在经历了几个可怕的星期后被释放，但这位无辜的伯爵就没那么幸运了：他在接下来的 16 年里都被关在监狱里。他设法通过玩战争游戏来消磨一些时间——并在塔的场地上建造了一条保龄球馆。但他也花时间思考科学，为了激发他对保龄球和台球的兴趣，他请哈里奥特解释碰撞力学。

Harriot came close to the idea, though, in his 1619 analysis of the mechanics of collisions. As well as war and war games, the game of billiards was in the air—at least for wealthy aristocrats such as Harriot’s patron Northumberland. Unfortunately, the earl soon landed in the Tower of London, a prisoner of King James I—both he and Harriot were arrested in 1605 when the Gunpowder Plot was foiled, because they had unknowingly dined with one of the Plotters just the day before. (The Plotters were disaffected Catholics chafing under the rule of the Protestant king and his parliament, and one of them happened to be Northumberland’s cousin.) Harriot was released after a few terrifying weeks, but the innocent earl was not so lucky: he spent the next sixteen years in prison. He managed to while away some of the time playing war games—and by having a bowling alley constructed in the Tower’s grounds. But he also spent his time thinking about science, and to build on his interest in bowls and billiards he asked Harriot to explain the mechanics of collisions.

哈里奥特的论文是对该主题的首次详细分析，也是牛顿之前最复杂的矢量分析之一。我在图 3.3中重现了哈里奥特的矢量图。球a和A的质量由圆圈的大小表示，它们从左侧相向而行，在中间（在b和B 处）相撞，然后反弹到虚线圆圈c和C表示的位置。我在尾注⁸中解释了他的计算，但只要看一下哈里奥特的图表，你就可以看出他使用平行四边形来分析他在此处展示的两种碰撞类型。尽管他的术语很过时——他谈论运动和动力而不是速度和力——但他掌握了现代平行四边形规则的精髓：大小和方向都是指定的，他有独立的部分，每个部分都“好像”另一个部分不存在一样，它们结合起来产生合力或合力。唯一的问题是，他没有发表它^。9

Harriot’s paper was the first detailed analysis of the topic—and one of the most sophisticated vectorial analyses before Newton. I’ve reproduced Harriot’s vectorial diagram in figure 3.3. The masses of the balls a and A are indicated by the size of his circles, and they move toward each other from the left, collide in the middle (at b and B), and bounce off to the positions denoted by the dotted circles c and C. I’ve explained his calculations in the endnote,⁸ but you can see just by looking at Harriot’s diagram that he used parallelograms to analyse the two types of collision he’s shown here. And although his terminology is old-fashioned—he speaks of motion and impetus instead of velocity and force—he has the essence of the modern parallelogram rule: both magnitudes and directions are specified, and he has independent components, each acting “as if ” the other weren’t there, which combine to give the resultant motion or force. The only trouble is, he didn’t publish it.⁹

不过，他的同胞约翰·沃利斯可能见过它——他肯定看过哈里奥特的一些手稿，以及他死后出版的代数教科书《实践》。正如我所提到的，沃利斯尤其受到哈里奥特的完全符号代数符号的启发，但他还探讨了碰撞的主题——牛顿认为他在这方面的工作是《自然哲学的数学原理》的另一个重要影响。沃利斯在 1671 年的一篇力学论文中清楚地阐述了平行四边形规则，在 1687 年《自然哲学的数学原理》出版之前的几十年里，其他几个人也这样做了——包括多才多艺的费马。（和哈里奥特的代数先驱韦达一样，费马的日常工作是法律，而不是数学。）¹⁰

His countryman John Wallis may have seen it, though—he’d certainly seen some of Harriot’s manuscripts, as well as his posthumous algebra textbook Praxis. Wallis was inspired especially by Harriot’s fully symbolic algebraic notation, as I’ve mentioned, but he also tackled the topic of collisions—Newton cited his work on this as another important influence on Principia. Wallis clearly stated the parallelogram rule in a 1671 paper on mechanics, and, in the decades leading up to 1687 when Principia was published, several others did so, too—including the remarkably versatile Fermat. (Like Harriot’s algebraic forerunner Viète, Fermat’s day job was in law, not mathematics.)¹⁰

你可能会认为，随着《数学原理》中平行四边形规则的成熟，以及牛顿对力、加速度和速度的矢量定义，正是他引发了对完整的数学矢量分析的探索——但事实并非如此，至少不是直接的。他的工作后来确实有所贡献，展示了矢量在物理学中的用处，但那是在第一个矢量代数从一个令人惊讶的、完全不同的方向出现之后。

You might think that, with the parallelogram rule coming of age in Principia, and with Newton’s vectorial definitions of force, acceleration, and velocity, it was he who sparked the search for a complete, mathematical vector analysis—but it wasn’t, at least not directly. His work did contribute later, by showing how useful vectors would be in physics, but that was after the first vector algebra had appeared from a surprising and completely different direction.

欧拉还给了我们指数（或欧拉）数的符号e——我在下面对其进行了定义——他还推广了威廉·琼斯的符号π。但就像琼斯的π一样，欧拉的i直到几十年后另一位杰出数学家——德国数学家卡尔·弗里德里希·高斯——才真正流行起来。高斯还创造了“复数”一词，用于表示实数和虚数的混合，我们今天将其写为a + ib或a + bi —— i的书写顺序无关紧要。但“复数”一词也没有立即流行起来。所有这些都表明创新思想的演变很缓慢——如果你想获得一些迟来的荣誉，发表论文是多么重要！

Euler also gave us the symbol e for the exponential (or Euler) number—which I’ve defined below—and he popularised William Jones’s symbol π. But just as with Jones’s π, Euler’s i didn’t really take off until another standout mathematician used it several decades later—the German Carl Friedrich Gauss. Gauss also coined the term “complex number,” for the mix of real and imaginary numbers we write today as a + ib or a + bi— the order in which you write the i doesn’t matter. But the term “complex” didn’t catch on straightaway, either. All of which goes to show the slow evolution of innovative ideas—and how crucial it is to publish if you want to get some belated credit!

顺便说一句，高斯还是自学成才的女性先驱索菲·热尔曼的不知情的导师。和他那个时代的大多数男人一样，高斯认为女性无法掌握高等数学，而热尔曼曾谨慎地用勒布朗先生的笔名给他写信，谈到了证明费马大定理的想法。当她最终透露自己是女性时，他感到惊讶，也非常钦佩。

Incidentally, Gauss was also the unknowing mentor of the self-taught female trailblazer Sophie Germain. Like most men of his time, Gauss believed women were incapable of higher mathematics, and Germain had prudently written to him—about an idea for proving Fermat’s last theorem—using the pseudonym Monsieur Le Blanc. When she finally revealed that she was a woman, he was amazed, and very impressed.

但是i和向量有什么关系呢？它取决于如何表示信息的问题，以及遵循明确数学规则的哪种数值结构有资格成为数学研究的对象的相关问题。

But what has i to do with vectors? It hinges on the question of how to represent information—and the related question of what kind of numerical constructs qualify as objects of mathematical study, by obeying clear mathematical rules.

就i而言，如果数学家们要认真对待这个数字，他们该如何思考呢？他们花了很长时间才将数轴延伸到零的左边来表示负数——这一看似简单的步骤使得人们能够将负数简单地视为正数的对应物，并且同样真实。沃利斯迈出了这一重大的一步——尽管他也认为负数必须大于无穷大，这表明即使是最优秀的数学家也很难掌握负数（更不用说无穷大了）。沃利斯也是第一个尝试对复数做类似的事情的人，但他没有成功^。11

In the case of i, if mathematicians were going to take this number seriously, how were they to think about it? It had taken long enough to extend the number line to the left of zero to represent negative numbers—a seemingly simple step that made it possible to think of negative numbers simply as counterparts to positive ones, and just as real. It was Wallis who took this momentous step—although he also thought that negative numbers must be larger than infinity, which shows how difficult it was for even the best mathematicians to get a handle on negative numbers (let alone infinity). It was also Wallis who first tried to do something similar for complex numbers, but he wasn’t successful.¹¹

这个表示问题又花了一个世纪才得到解决，并无意中为发现向量开辟了道路。与此同时，数学家需要更多地了解复数的数学性质。当然，它们是作为二次和三次方程的解而出现的，但它们在数学中是否发挥了更广泛的作用——这个角色使得必须找到一种方法将它们纳入扩展的数字体系中？

It would take another century for this representation problem to be solved, and to inadvertently open the way for the discovery of vectors. Meantime, mathematicians needed to find out more about the mathematical nature of complex numbers. Sure, they came up as solutions of quadratic and cubic equations, but did they play a broader role in mathematics—a role that would make it imperative to find a way to incorporate them into an expanded system of numbers?

欧拉是最早找到这个问题答案的人之一。他是历史上最多产的数学家之一，他拥有惊人的记忆力和非凡的奉献精神，甚至失明也无法阻止他。1735 年，年仅 28 岁的他首先失去了右眼视力，59 岁时几乎失明，但在秘书以及子孙的帮助下，他继续研究，直到 76 岁时突然去世。他曾是令人敬畏的约翰·伯努利（或让·伯努利）的学生——伯努利是莱布尼茨的坚定捍卫者和牛顿的鞭策者——他是第一批用符号微积分明确改写牛顿定律的人之一。他于 1727 年开始与伯努利一起研究复数，但他在这个主题上的工作真正开始形成是在 18 世纪 40 年代，当时他将i与“圆”函数 sin θ 和 cos θ 联系起来。 （请耐心听我说完，因为圆与旋转相关，掌握旋转代数将引导汉密尔顿创建矢量。但首先他需要理解旋转和i之间的联系。）

One of the first to find answers to this question was Euler. He was one of the most prolific mathematicians in history, and he had such a phenomenal memory and such extraordinary dedication that not even blindness stopped him. He lost sight first in his right eye, in 1735 when he was only twenty-eight, and became almost blind at fifty-nine, but with the help of a secretary and his children and grandchildren, he continued researching until he died suddenly at seventy-six. He had been a student of the formidable Johann (or Jean) Bernoulli—the fierce defender of Leibniz and scourge of Newton—and he was one of the first to rewrite Newton’s laws explicitly in terms of symbolic calculus. He began working on complex numbers with Bernoulli in 1727, but his work on this topic really began to come together in the 1740s, when he connected i with the “circular” functions sin θ and cos θ. (Bear with me here, because circles are related to rotations, and coming to grips with the algebra of rotations is what will lead Hamilton to create vectors. But first he will need to understand the connection between rotations and i.)

如果你将 sinθ 和 cosθ 表示在一个半径为 1 的圆上，如图 3.4所示，那么利用毕达哥拉斯定理，你可以得到有用的恒等式

If you represent sinθ and cosθ on a circle with radius 1, as in figure 3.4, then, using Pythagoras’s theorem, you get the helpful identity

(余弦θ) ²⁺ (罪θ) ² =1。

(cos θ)² + (sin θ)² = 1.

欧拉意识到可以将此方程分解为：

Euler realised that you could factorise this equation as:

(cos θ + i sin θ)(cos θ − i sin θ) = 1。

(cos θ + i sin θ)(cos θ − i sin θ) = 1.

从那里开始，他使用e的一个不成熟的定义（你可以在尾注¹²中看到），得出了现在被称为欧拉公式的公式：

From there, using a fledgling definition of e (which you can see in the endnote¹²), he came up with what is now known as Euler’s formula:

e ^{i θ} = cos θ + i sin θ。

e^iθ = cos θ + i sin θ.

（如果你像欧拉一样知道 sinθ、cos θ 和e ^θ的泰勒级数，那么这一点就很容易理解了，但这些细节对于我们的故事并不重要。）

(It follows easily if you know, as Euler did, the Taylor series for sinθ, cos θ, and e^θ, but these details are not important for our story.)

欧拉的这个非凡公式可以做很多事情。它能让你做纯数学的事情，比如求平方、立方和更高的数复数根在第 1 章中提到过。而且它在实际应用中也非常有用，例如模拟电路和其他波状现象——包括量子力学，如薛定谔方程（它控制着表示移动亚原子粒子（如电子）可能状态的“波函数”）。这是因为从数学上讲，波是周期性的循环，例如当你将 θ 旋转一整圈时，你会从 cosθ 和 sin θ 得到这样的循环（如图3.5所示）。

You can do a lot with Euler’s remarkable formula. It enables you to do pure mathematical things such as finding the square-, cube-, and higher roots of complex numbers mentioned in chapter 1. And it turns out to be incredibly useful in practical applications, too, such as modeling electric circuits and other wave-like phenomena—including in quantum mechanics, as in Schrödinger’s equation (which governs the “wave function” representing the possible state of a moving subatomic particle such as an electron). That’s because mathematically, waves are periodic cycles, such as you get from cosθ and sin θ when you turn θ through a whole circle (as in fig. 3.5).

绕圆旋转也将成为理解实数和虚数之间关系的关键。事实上，由于π在圆的周长和面积公式中起着如此重要的作用，你只需在欧拉公式中选择θ=π，就能看出圆和虚数之间存在某种联系。你会得到一个经常被选为数学中最漂亮的方程：

Rotations around a circle will also turn out to be the key to understanding the relationship between real and imaginary numbers. In fact, since π plays such an important role in the formulae for the circumference and area of a circle, you can already see there’s some sort of connection between circles and imaginary numbers simply by choosing θ = π in Euler’s formula. You get what is often voted the most beautiful equation in mathematics:

e^我π + 1 = 0。

e^iπ + 1 = 0.

它之所以美丽，是因为它简洁优雅，却又意义深远。它只有 7 个符号，而且它们以一种完全出乎意料的方式相互关联：除了基本运算符号 + 和 = 之外，还有自然对数的底数 e、所有圆心的神秘数字 π、最重要的整数 0 和 1，以及虚数i 。谁会想到所有这些不同类型的数字都是相关的？这是一个奇迹般的等式，尽管欧拉从未真正写下来。但由于它是欧拉公式的一个明显进步，并且没有记录表明谁是第一个提出它的人，所以今天它被合理地称为欧拉恒等式^。13

It’s beautiful because it is elegantly simple, yet profound. It has just seven symbols, and they are interconnected in an entirely unexpected way: aside from the basic operation signs + and = there is e, the base of the natural logarithms; π, the mysterious number at the heart of all circles; 0 and 1, the most important integers; and the imaginary number i. Who’d have thought that all these different types of number were related? It is a marvel of an equation, although Euler never actually wrote it down. But since it is such an obvious step from Euler’s formula, and since there is no record of who was the first to take it, today it is justifiably called Euler’s identity.¹³

然而，欧拉仍然对复数的性质感到困惑，而且他也没有在用某种“数轴”表示复数方面取得任何进展。所以，没有人真正知道所有这些数字是如何联系在一起的。

Euler was still confused about the nature of complex numbers, though, and he didn’t make any progress on representing them on some sort of “number line,” either. So, no one really knew how all these numbers were connected.

尽管如此，欧拉还是继续以代数方式使用复数，乍一看可能有点令人惊讶：试图证明费马大定理。或者，我应该说，试图证明它的一个特例——除了 0 和 1 之外，没有整数x、y、z使得x ³ + y ³ = z ³。由于这是一个三次方程，因此可以尝试使用卡尔达诺算法来“解决”它，该算法习惯于求出复数。其思想是，当你发现解决方案，然后证明它与问题固有的假设相矛盾——例如x、y、z必须是整数！这个过程称为矛盾证明，在纯数学中非常有用。欧拉采用了与邦贝利类似的方法，通过探索形式为$\mathrm{一个}+b\sqrt{-3}$（那是，$\mathrm{一个}+我b\sqrt{3}$）。

Still, Euler did go on to use complex numbers algebraically, in what might at first seem a surprising way: trying to prove Fermat’s last theorem. Or, should I say, trying to prove a special case of it—that aside from 0 and 1 there are no integers x, y, z such that x³ + y³ = z³. Since this is a cubic equation, it is possible to try to “solve” it with Cardano’s algorithm, which has a habit of turning up complex numbers. The idea is that when you find a solution, you then prove that it contradicts the assumptions inherent in the problem—such as the fact that x, y, z have to be integers! This process is called proof by contradiction, and it is very powerful in pure mathematics. Euler took an approach similar to that of Bombelli, by exploring cubes and cube roots of numbers of the form $a+b\sqrt{-3}$ (that is, $a+ib\sqrt{3}$).

费马大定理引起了所有人的兴趣，该定理指出，如果n是大于 2 的整数，则不存在非平凡的x、y、z使得x ⁿ + y ⁿ = z ⁿ。这个定理之所以有趣，是因为找到x ² + y ² = z ²的解并不困难——古老的普林普顿 322 石板上有完整的解列表。17 世纪 30 年代，费马曾写下著名的笔记，表示他已经找到了关于他猜想的真正绝妙的普遍证明——但如果他真的找到了，也没人能找到。他确实在n = 4 时给出了证明，而欧拉在n = 3 时给出了证明，尽管晚了一个世纪^。14

Everyone was intrigued by Fermat’s last theorem, which says that if n is an integer larger than 2, there are no nontrivial x, y, z such that xⁿ + yⁿ = zⁿ. It was intriguing because there was no problem in finding solutions to x² + y² = z²—the ancient Plimpton 322 tablet has a whole list of them. In the 1630s, Fermat had famously scribbled a note indicating that he had found a truly marvelous general proof of his conjecture—but if he did, no one has been able to find it. He did give a proof when n = 4, and Euler’s proof for n = 3 was the next step, albeit a century later.¹⁴

在接下来的两个世纪里，许多数学家（包括杰曼）都贡献了特殊情况的证明，直到 20 世纪 90 年代，安德鲁·怀尔斯在新思想和强大计算机能力的帮助下证明了整个证明。但欧拉是证明之路上的潮流引领者，这意味着他使用a + ib形式的代数数也引起了人们的注意，尽管其他人已经做了类似的事情——包括法国人让·勒朗·达朗贝尔，他对欧拉没有承认他感到很恼火。欧拉非常擅长发展他人的新兴思想并将其转化为最终形式，就像他对牛顿定律所做的那样——但他对承认来源相当粗心。也许他阅读范围太广，将思想带入了如此深奥的领域，以至于他完全忘记了自己在哪里读过什么^。15

Over yet two more centuries, many mathematicians—including Germain—contributed proofs of special cases, until Andrew Wiles proved the whole thing in the 1990s, with the help of new ideas and massive computer power. But Euler was the trendsetter on the road to a proof, and this meant that his use of algebraic numbers of the form a + ib also got noticed, although others had already done something similar—including Frenchman Jean Le Rond d’Alembert, who was miffed that Euler hadn’t acknowledged him. Euler was terrific at developing others’ fledgling ideas and putting them in their definitive form, the way he did with Newton’s laws—but he was rather careless about acknowledging his sources. Perhaps he read so widely and took ideas into such deeper territory that he simply lost track of where he’d read what.¹⁵

欧拉没有在复数上取得进一步进展，但他在这两个项目中取得的成就意义重大——将i与圆函数联系起来，并将a + ib视为代数值，如果分别加减实部和虚部，其行为就像普通数字一样。但半个世纪后，威廉·罗文·汉密尔顿 (William Rowan Hamilton) 却在最后一点上遇到了麻烦，因为他认为这就像把苹果和橘子加在一起一样^。16

Euler didn’t go any further with complex numbers, but what he achieved with these two projects was significant—linking i with the circular functions, and treating a + ib as an algebraic quantity that behaved like an ordinary number if you added and subtracted the real and imaginary parts separately. But it was this last bit that William Rowan Hamilton had trouble with, half a century later, for he thought it was like adding apples and oranges.¹⁶

当然，自欧拉时代以来已经发生了很多事情。特别是数学家终于找到了一种方法来表示这些奇怪而复杂的图 3.6 中的复平面表示法是一种将复平面表示为某种数轴上的数字的表示法。它是几何的而非代数的，但它非常直观，并且在很大程度上帮助数学家们放心地将这些混合体视为数字。在沃利斯的尝试一个多世纪后，这个解决方案终于应运而生，因为几位数学家独立提出了类似于图 3.6中的现代表示法的想法。其中最著名的是 1799 年的挪威测量员卡斯帕·韦塞尔 (Caspar Wessel)，尽管他的开创性努力在接下来的一个世纪里基本上被埋没了；大约在同一时间的高斯，尽管他直到 1831 年才发表论文；阿尔冈 (Argand)（其传记不确定，但可能是出生于瑞士的巴黎书店经理让·罗伯特·阿尔冈 (Jean Robert Argand)）于 1806 年提出——无论他是谁，我们通常将图 3.6中的“复平面”称为“阿尔冈平面”以纪念他——以及 1828 年的英国人约翰·沃伦 (John Warren)。

Of course, a lot had happened since Euler’s time. In particular, mathematicians had finally found a way to represent these strange, complex numbers on some sort of number line—or rather, as it turned out, a plane. It was geometric rather than algebraic, but it was brilliantly intuitive, and it went a long way toward helping mathematicians feel secure in treating these hybrid beasts as numbers. More than a century after Wallis’s attempt, it was a solution whose time had come, for several mathematicians independently came up with ideas similar to the modern representation in figure 3.6. The most notable of them are the Norwegian surveyor Caspar Wessel in 1799, although his pioneering effort lay largely buried for the next century; Gauss around the same time, although he didn’t publish till 1831; Argand (whose biography is uncertain but who was probably the Swiss-born Parisian bookstore manager Jean Robert Argand), in 1806— and whoever he was, we often call the “complex plane” in figure 3.6 the “Argand plane” in his honour—and Englishman John Warren in 1828.

最后，我一直在谈论的一些不同的想法开始结合在一起：分量、圆、虚数和矢量。

And finally, some of the disparate ideas I’ve been talking about began to come together: components, circles, imaginary numbers, and a glimpse of vectors.

首先，如果你从图 3.4中取出单位圆的概念，并将其叠加到图 3.6中的复平面上，就会得到图 3.7——一个以实数线和虚数线原点为中心的单位圆。数字 +1 对应于 θ = 0；i对应于$\text{θ}=\frac{\pi }{2}$弧度或 90°；−1 对应于 θ = π 或 180°，−i对应于$\text{θ}=\frac{3\pi }{2}$或 270°。（或者你可以顺时针旋转，这样 − i对应于$\text{θ}=-\frac{\pi }{2}$或 −90° 等。）但最精彩的部分在于，如果你在脑海中沿着圆从点 1 移动到点i，那么你就旋转了 90°。换句话说，旋转 90° 会将数字 1 变成数字i —就像你写 1 × i = i一样。就好像将实数 1 旋转 90° 相当于将其乘以i一样。现在将i处的点旋转90°：你会得到 −1，就像你说i ² = −1一样。乘以虚数确实似乎只不过是一个简单的旋转！所以，复平面的奇妙之处在于它展示了虚数如何与实数相关联：通过几何旋转。

First, if you take the idea of the unit circle from figure 3.4 and superimpose it on the complex plane in figure 3.6, you get figure 3.7—a unit circle centred at the origin of the real and imaginary number lines. The number +1 corresponds to θ = 0; i corresponds to $\text{θ}=\frac{\pi }{2}$ radians or 90°; −1 corresponds to θ = π or 180°, and −i corresponds to $\text{θ}=\frac{3\pi }{2}$ or 270°. (Or you can go clockwise so that −i corresponds to $\text{θ}=-\frac{\pi }{2}$ or −90°, and so on.) But here’s the fascinating part. If you mentally move along the circle from the point 1 to the point i, you’ve rotated through 90°. In other words, a rotation of 90° takes the number 1 to the number i—just as when you write 1 × i = i. It’s as if the act of rotating the real number 1 through 90° is equivalent to multiplying it by i. Now rotate the point at i by 90°: you get −1, just as you do when you say i² = −1. Multiplication by an imaginary number does indeed seem to be nothing more than a simple rotation! So, the wonderful thing about the complex plane is that it shows just how imaginary numbers can be related to real ones: via geometric rotations.

此外，一个由实部和虚部组成的复数，表示为a + ib （或a + bi，随你喜欢），可以用几何形式表示为这个复平面上的一个点( a, b )（如图 3.6所示）。如果你事后再想象从原点到这个点的箭头，你会看到它有一个实部和一个虚部。它显然也有一个方向，以及一个通过毕达哥拉斯定理从分量计算出来的量级。（实际上，复数有一个“模数”而不是量级：复数a + bi的模数定义为$\sqrt{{\mathrm{一个}}^2+b^2}$，类似于后来出现的矢量的大小。）因此，你可以看到，这种表示复数的几何方法的发明者——阿尔冈、高斯、韦塞尔和沃伦——正在接近矢量的概念，就像牛顿所做的那样。牛顿从物理学的角度来处理这个问题，强调力等物理量的大小和方向，而这些十九世纪早期的数学家则从数论的角度来处理这个问题。汉密尔顿需要明确定义矢量，并将这两种方法结合起来。

What’s more, a complex number, made of real and imaginary parts and expressed as a + ib (or a + bi, whichever you prefer), can be represented geometrically as a point (a, b) in this complex plane (as in fig. 3.6). And if you imagine, with hindsight, an arrow from the origin to this point, you see that it has a real and an imaginary component. It obviously has a direction, too, and a magnitude calculated from the components via Pythagoras’s theorem. (Actually, a complex number has a “modulus” rather than a magnitude: the modulus of the complex number a + bi is defined to be $\sqrt{a^2+b^2}$, analogous to the magnitude of a vector, which came later.) So, you can see that the inventors of this geometrical way of representing complex numbers—Argand, Gauss, Wessel, and Warren—were closing in on the idea of a vector, just as Newton had done. Newton had approached the problem from the perspective of physics, emphasising the magnitude and direction of physical quantities such as force, while these early nineteenthcentury mathematicians were coming at it from a number theory point of view. It would take Hamilton to define vectors explicitly, and to bring the two approaches together.

他用一种相当聪明的方法解决了这个难题。如果你先取任意两个混合数，然后像对待普通二项式一样对它们进行加法和乘法，但i ² = −1，你会得到以下结果：

He got around the difficulty in a rather clever way. If you start by taking any two of these hybrid numbers and adding and multiplying them as if they were ordinary binomial expressions, except that i² = −1, you get this:

（a + ib）+（c + id）=（a + c）+ i（b + d），

（a + ib）（c + id）=（ac − bd）+ i（ad + bc）。

(a + ib) + (c + id) = (a + c) + i(b + d),

(a + ib)(c + id) = (ac − bd) + i(ad + bc).

如今，我们在数学课上经常这样做，欧拉、高斯等人也这样做过，尽管当时这似乎有点敷衍，因为当时还没有人对这种算术感到满意——或者我应该说，对这种数字感到满意。正如高斯告诉天文学家彼得·汉森的那样，“$\sqrt{-1}$生动地展现在我的灵魂面前，但用语言来表达却非常困难，因为语言只能给出一个悬浮在空中的图像。”汉密尔顿的朋友奥古斯都·德·摩根声称“展示了这个符号$\sqrt{-1}$毫无意义，或者说自相矛盾、荒谬可笑。” ¹⁷

We do it all the time in maths classes today, and Euler, Gauss, and others had done it, too, although back then it still seemed somehow like fudging, because no one was yet comfortable with this kind of arithmetic—or, should I say, with this kind of number. As Gauss told the astronomer Peter Hansen, “The true meaning of $\sqrt{-1}$ reveals itself vividly before my soul, but it will be very difficult to express it in words, which can give only an image suspended in air.” And Hamilton’s friend Augustus De Morgan claimed to have “shown the symbol $\sqrt{-1}$ to be void of meaning, or rather self-contradictory and absurd.”¹⁷

因此，汉密尔顿说，为什么不定义复数时去掉分散注意力的i，即苹果中散落的橘子，同时仍然保留相同的加法和乘法规则呢？为什么不把复数完全看作实数的“对”（或现代数学家所说的“有序对”）—— （ a ，b），（c，d）等等——然后定义加法和乘法，使它们遵循相同的模式：

So, Hamilton said, why not define complex numbers without the distracting i, the stray orange among the apples, yet still keep the same rules of addition and multiplication. Why not think of complex numbers entirely in terms of “couples” (or what modern mathematicians call “ordered pairs”) of real numbers—(a, b), (c, d), and so on—and then define addition and multiplication so that they followed the same pattern:

（a，b）+（c，d）=（a + c，b + d），

（a，b）×（c，d）=（ac − bd，ad + bc）。

(a, b) + (c, d) = (a + c, b + d),

(a, b) × (c, d) = (ac − bd, ad + bc).

早期数学家，尤其是高斯，已经接近这个想法，但汉密尔顿将其阐述得很清楚。起初并没有取得多大成功。几千年来，乘法只是一个数字乘以另一个数字，所以汉密尔顿的乘法，混合了乘法、加法和减法，是非常不同的东西。

Earlier mathematicians, especially Gauss, had come close to this idea, but Hamilton spelled it out clearly. Not that it met with much success at first. For thousands of years multiplication had been just one number multiplied by another, so Hamilton’s multiplication of couples, with its jumbled-up mixture of multiplications, additions, and subtractions, was something very different.

但时机已到，因为他的英国同事德·摩根和乔治·皮科克已经开始探索完全符号代数的概念——从哈里奥特和沃利斯转向代数的概念，这种代数不仅脱离几何，也脱离算术。换句话说，代数用符号来定义，这些符号不必有任何具体的含义，但必须遵循明确既定的规则——就像汉密尔顿的“配对”一样。皮科克和德·摩根首先正式表达了普通代数的规则，尽管这些规则已经被直观地使用了数百年甚至数千年。毕竟，三个苹果加两个还是两个苹果加三个都无关紧要，所以你会认为它几乎不需要称为“加法交换律”。当然，除非你想把代数从数值转移到纯粹抽象符号和运算的科学，就像皮科克和他的前学生德·摩根所做的那样。

But its time was near, for his English colleagues De Morgan and George Peacock had already begun exploring the idea of a completely symbolic algebra—moving on from Harriot and Wallis to an idea of algebra divorced not just from geometry but from arithmetic, too. In other words, algebra defined in terms of symbols that didn’t have to mean anything concrete, but which did have to obey clearly established rules—just as Hamilton’s “couples” do. Peacock and De Morgan had begun by expressing the rules of ordinary algebra formally, even though they’d been used intuitively for hundreds, perhaps thousands of years. After all, it doesn’t matter if you add three apples to two or two apples to three, so you’d think it hardly needed stating as the “commutative law for addition.” Unless, of course, you wanted to move algebra beyond numerical quantities to a science of purely abstract symbols and operations, as Peacock and his former student De Morgan did.

皮科克对符号系统的兴趣最初是在微积分的背景下形成的。在第二章中，我提到了与牛顿的点符号相比，莱布尼茨微分dx、dy等富有启发性的威力，而皮科克曾是一群年轻的剑桥学者中的一员，其中包括因原型计算机而出名的查尔斯·巴贝奇和约翰·赫歇尔，后者是发现天王星的著名天文学家的儿子，也是一位著名的天文学家。他们鼓吹课程改革，用莱布尼茨的微积分取代牛顿的微积分。正如巴贝奇顽皮地说的那样——既是对正式宗教的嘲讽，也是对剑桥大学对其昔日学生牛顿的敬畏——这些特立独行的年轻人支持“纯粹的d主义原则，而不是大学的点时代” ^。18

Peacock’s interest in symbolism had initially taken shape in the context of calculus. In chapter 2, I mentioned the power of the suggestive Leibnizian differentials dx, dy, and so on, compared with Newton’s dot notation, and Peacock had been part of a group of young Cambridge academics—including Charles Babbage of prototype-computer fame, and John Herschel, son of the famous astronomer who discovered Uranus, and a significant astronomer in his own right—who’d agitated for a curriculum reform to replace Newton’s calculus with Leibniz’s. As Babbage impishly put it—punning on both formal religion and the awe in which Cambridge still held its former student Newton—these young mavericks supported “the principles of pure d-ism as opposed to the dot-age of the university.”¹⁸

至于纯符号代数，汉密尔顿本人最初横跨两个阵营。他同意复平面上的几何表示是说明性的，而不是从根本上讲是数学的。但他并不是一个纯粹的符号主义者，因为他认为代数应该是比一套规则更具体的东西。他甚至想知道这两个阵营是否像空间和时间一样相互联系。德国哲学家伊曼纽尔康德曾说过，我们的存在取决于预先存在的空间和时间，几何学是空间表示的基本语言——所以，汉密尔顿想，也许代数是时间的科学？毕竟，他说，我们标记时间和空间的方式是“思想变成事物，精神穿上身体，心灵的行为和激情被赋予外在存在，我们从远处观察自己”的手段^。19

As for purely symbolic algebra, Hamilton himself initially straddled both camps. He agreed that geometrical representations on the complex plane were illustrative rather than fundamentally mathematical. But he wasn’t a pure symbolist, because he felt that algebra should be about something more tangible than a set of rules. He’d even wondered if the two camps were linked like space and time. The German philosopher Immanuel Kant had said that our existence depends on preexisting space and time, and that geometry is the fundamental language of spatial representation—so, thought Hamilton, perhaps algebra was the science of time? After all, he said, the way we mark time and space is the means “by which thoughts become things, and spirit puts on body, and the act and passion of mind are clothed with an outward existence, and we behold ourselves from afar.”¹⁹

抛开诗歌不谈——汉密尔顿从年轻时就开始写诗——如今我们已经习惯了用代数方法处理任意的x和y或a和b或我们选择的任何符号，他的时间代数听起来很奇怪。他的符号主义同事也不喜欢它。但正如莱布尼茨和其他人很久以前指出的那样，我们的时间概念（以及空间概念）是相互关联的——这件事发生在那件事之前，这件事正在发生，等等。爱因斯坦将这个想法推向了合乎逻辑的结论，在这个过程中提出了一些关于时间变慢和“现在”本身是相对的奇怪新想法；与此同时，汉密尔顿意识到这些关系时间时刻之间的时间差可以用代数表达式来表达。他的重点是纯数学而不是物理学——否则他可能已经预见到了爱因斯坦；事实上，他引用了牛顿的话，牛顿在描述事物如何变化时，曾诉诸于增加的“时刻”和时间的“流动”，他提出了我们现在称之为导数的概念，但他当时称之为“流动数”。汉密尔顿将时间类比带入了更基本的领域，他建议，如果你把数字本身看作表示有向线段上“时刻”之间的心理“步骤”，你就可以“解释”算术定律作为一门时间科学。

Poetry aside—Hamilton had been writing poetry since his youth—his algebra of time sounds odd today, when we are so used to algebraically manipulating arbitrary x’s and y’s or a’s and b’s or whatever symbols we choose. His symbolist colleagues didn’t like it either. But as Leibniz and others had pointed out long ago, our notion of time (and of space, too) is relational—this happened before that, this is happening now, and so on. Einstein would take this idea to its logical conclusion, turning up in the process some bizarre new ideas about time slowing down and “now” itself being relative; in the meantime, Hamilton realised that these relations between moments of time could be expressed in terms of algebraic expressions. His focus was pure mathematics rather than physics—otherwise he might have anticipated Einstein; as it is, he cited Newton, who had appealed to “moments” of increase and the “flux” of time when describing how things change, in his concept of what we now call a derivative but which he had called a “fluxion.” Hamilton took the time analogy into more basic territory by suggesting that if you think of numbers themselves as representing the mental “steps” between “moments” on a directed line segment, you can “explain” the laws of arithmetic as a science of time.

这些“步骤”相当深奥，尽管后来他从时间中的瞬间转移到空间中的点，并将他的“步骤”定义为向量；无论如何，你可以通过一个例子来了解他的意思：令人困惑但似乎不可避免的事实是两个负数的乘积是正数，因此 (-3) × (-1) 等于 3。正如汉密尔顿所说，在他的哲学中，你可以用“两次连续的逆转恢复了步骤的方向”来解释这个奇怪的结果。这里的要点是方向在他的矢量概念中的重要性：将箭头的方向反转两次，你就会回到起点。这类似于说“我不会不去”——两个连续的“不”相互抵消，就像两个减号取消或“恢复”我们到正方向一样。当你将两个小于零的数字相乘但最终得到正数时，这当然有助于理解事情！²⁰

These “steps” were rather esoteric, although later he moved on from moments in time to points in space, and he identified his “steps” as vectors; either way, you can get his drift in an example: the confusing but seemingly inescapable fact that the product of two negative numbers is positive, so that (−3) × (−1), say, is equal to 3. As Hamilton put it, in his philosophy you could explain this bizarre result by saying that “two successive reversals restore the direction of a step.” The take-home message here is the importance of direction in his conception of a vector: reverse the direction of an arrow twice and you’re back where you started. It’s analogous to saying “I will not not go”—two successive “nots” cancel each other out, just as two minus signs cancel or “restore” us to the positive direction. It certainly helps make sense of things when you’re multiplying two numbers that are each less than zero and yet you end up with a positive!²⁰

汉密尔顿最终确实更接近于纯符号方法，今天看来，这在他对“耦合”算术的定义中显而易见。德摩根也几乎明白这一点。他“倾向于认为”，如果你抛开时间，只关注实数，汉密尔顿“可能最终”提供了一种通过符号代数而不是直观的几何类比来思考复数的方法^。21

Hamilton did eventually move closer to the purely symbolic approach that seems evident, today, in his definition of the arithmetic of “couples.” It had been almost evident to De Morgan, too. He was “inclined to think” that if you left time out of it and focussed simply on real numbers, Hamilton “may finally” have provided a way to think about complex numbers through symbolic algebra, not just intuitive geometric analogy.²¹

如今，我们很容易想知道为什么有人会担心负数和复数的特殊性以及代数规则——我们已经习惯了在需要的时候使用这些东西，而不用担心潜在的概念问题。但当然，这是因为一些最伟大的数学家确实担心正是这些事物，我们才有了现代数学和许多技术。汉密尔顿在 1837 年发表的关于复杂耦合的论文中，一开始就说他的重点是纯粹的数学。他当时并不知道，今天 NASA 会利用他的研究成果来帮助驱动其航天器！所以，即使他最初对时间的关注似乎是错误的，但正是这种打破常规的思维——对数字和算术基础的关注——最终让汉密尔顿发明了向量和四元数，而它们在今天有着如此多的实际应用。

Today, it’s easy to wonder why anyone would worry about the peculiarities of negative and complex numbers and the rules of algebra—we are so used to just getting on with it and using these things when we need to, without worrying about underlying conceptual problems. But, of course, it is because some of the greatest mathematical minds did worry about such things that we have both modern maths and much of our technology. When Hamilton introduced his paper on complex couples, which he published in 1837, he said right up front that his focus here was purely mathematical. Little did he know that today NASA would be using the ultimate fruits of his research to help drive its spacecraft! So even if his initial focus on time seems misguided, it was this kind of out-of-the-box thinking—this concern with the very foundations of number and arithmetic—that would ultimately lead Hamilton to the invention of vectors and quaternions, which have so many practical applications today.

值得注意的是，汉密尔顿的灵感来源不仅包括数学家，还包括康德等哲学家，以及作家和诗人：威廉·华兹华斯是他的好友，著名小说家和教育理论家玛丽亚·埃奇沃思也是他的好友，浪漫主义者弗朗西斯·博福特·埃奇沃思（玛丽亚同父异母的弟弟）和塞缪尔·泰勒·柯勒律治也是他的好友。华兹华斯说，汉密尔顿和柯勒律治是他见过的最出色、最有天赋的两个人。玛丽亚·埃奇沃思第一次见到当时只有 19 岁的汉密尔顿时，她认为他“既有真正的天才的朴素，又有真正的天才的坦率” ^。22

It is telling that Hamilton’s sources of inspiration included not just mathematicians but also philosophers such as Kant—and writers and poets, too: William Wordsworth was a good friend, and so were the famous novelist and educational theorist Maria Edgeworth along with the Romantics Francis Beaufort Edgeworth (Maria’s younger half-brother) and Samuel Taylor Coleridge. Wordsworth said Hamilton and Coleridge were the two most wonderful and gifted men he had ever met. And when Maria Edgeworth first met Hamilton, who was then only nineteen, she thought he had “both the simplicity and the candour which make a true genius.”²²

这种兼收并蓄的影响在十九世纪并不像今天这样罕见，因为如今数学和科学一样，已经变得如此复杂和专业化。但即使在那个年代，汉密尔顿也不是一个普通人。

Such eclectic influences were not as unusual in the nineteenth century as they are today, now that mathematics, like science, has become so sophisticated and specialised. But Hamilton was no ordinary man, even in those days.

虽然詹姆斯叔叔对威廉同样非凡的计算天赋并不十分在行，但他还是尽力为威廉寻找合适的教科书。年轻的威廉因此能够利用他的法语技能，通过法语教科书自学微积分——所以他学习的是莱布尼茨微积分，而不是牛顿形式——并且在十六岁时，他开始学习皮埃尔-西蒙·拉普拉斯的《天体力学》。这是牛顿《自然哲学的数学原理》的一次重大更新，吸收了过去一个世纪数学天文学的所有进展。拉普拉斯是一位伟大的数学家，但十几岁的汉密尔顿发现他推理中的一个缺陷——恰好是关于合成力的平行四边形规则的。显然，他已经在开拓矢量的道路上前进了。

Uncle James wasn’t quite so able when it came to William’s equally prodigious talent for calculating, but he did his best to find suitable textbooks for him. Young William was thereby able to use his French skills to teach himself calculus from a French textbook—so he learned Leibnizian calculus rather than Newton’s form—and at sixteen he began to study Pierre-Simon Laplace’s Traité de Mécanique Celeste (Treatise on Celestial Mechanics). This was a monumental update of Newton’s Principia, incorporating all the advances in mathematical astronomy that had taken place in the intervening century. Laplace was a great mathematician, but the teenaged Hamilton spotted a flaw in his reasoning—about, as it happens, the parallelogram rule for composing forces. Evidently, he was already on the road to pioneering vectors.

那是 1821 年。与此同时，一位自学成才的苏格兰女性也在私下研究《天体力学》，十年后，她以教科书《天体力学》震惊英国数学界。她就是玛丽·萨默维尔——我们之前见过她，当时她九十多岁，正在研究汉密尔顿的四元数——她的书是拉普拉斯五卷本前两卷的扩展和详尽的 610 页英文版，外加一份 70 页的“初步论文”，为非专业人士提供概述。玛丽亚·埃奇沃思读了这篇概述后被迷住了：她发现她朋友萨默维尔的写作风格简单，非常适合“科学的崇高”。至于学术界，乔治·皮科克是剑桥大学数学家之一，他对整本书印象深刻，以至于他们用它作为高级天文学学生的教科书。 （诚​​然，书中对莱布尼茨微积分的使用也推动了皮科克的课程改革——但这本书太优秀了，以至于在下个世纪一直是标准教材。）萨默维尔非常激动——尤其是当评论家评论说，一位自学成才的女性写了一本很少有男性能理解的书，这是多么了不起。但她也非常愤怒，因为女性仍然被剥夺了接受良好教育的权利——在 19 世纪 60 年代末，她是第一个签署约翰·斯图尔特·密尔（最终未获成功）的女性投票权请愿书的人^。23

This was in 1821. At the same time, a self-taught Scottish woman was also privately studying Mécanique Celeste, and ten years later she would astonish the British mathematical community with her textbook Mechanism of the Heavens. She was Mary Somerville—we met her earlier, when she was studying Hamilton’s quaternions in her nineties—and her book was an expanded, explicated 610-page English version of the first two volumes of Laplace’s five-volume tome, plus a 70-page “preliminary dissertation” giving an overview for the nonspecialist. Maria Edgeworth read the overview and was entranced: she found that the simplicity of her friend Somerville’s writing superbly suited “the scientific sublime.” As for the academics, George Peacock was one of the Cambridge mathematicians who were impressed with the whole book—so much so that they used it as a text for their advanced astronomy students. (True, its use of Leibnizian calculus also helped further Peacock’s curriculum reform—but the book was so good that it remained the standard text for the next century.) Somerville was thrilled—especially when reviewers commented on how extraordinary it was that a self-taught woman had written a book that few men could understand. But she was furious, too, because women were still denied a decent education—and in the late 1860s she would be the first to sign John Stuart Mill’s (ultimately unsuccessful) petition for votes for women.²³

相比之下，汉密尔顿确实上了大学——就读于都柏林圣三一学院，那里的古典文学和数学教学都非常出色。他的表现非常出色，不到 22 岁就被任命为爱尔兰天文学教授和皇家天文学家。几年后，正如我所提到的，他因对圆锥折射的数学预测而闻名。然后，还在 20 多岁时，他就提出了一种新的动力学解释，这种解释在物理学的许多领域都具有重要意义——例如，在运动物体系统的力学中，以及在描述电子 (量子) 力学的薛定谔方程中。1833 年，当他向爱尔兰皇家科学院提交关于复数“对”的初步想法时，他才 28 岁，30 岁时，他因对数学的贡献而被授予爵士称号。

Hamilton, by contrast, did go to university—to Trinity College Dublin, where the teaching was excellent, in both classics and mathematics. He did so brilliantly that he was made professor of astronomy and royal astronomer of Ireland when he was not quite twenty-two. A few years later he became famous for his mathematical prediction of conical refraction, as I’ve mentioned. Then, still in his twenties, he formulated a new interpretation of dynamics that would become significant in many areas of physics—in the mechanics of systems of moving objects, for instance, and in Schrödinger’s equation describing the (quantum) mechanics of an electron. He was still only twenty-eight when he presented his initial ideas on complex number “couples” to the Royal Irish Academy in 1833, and he was knighted at thirty for his services to mathematics.

1837 年底，32 岁的汉密尔顿当选为爱尔兰皇家学院院长，他做的第一件事就是向刚满 70 岁的玛丽亚·埃奇沃思寻求建议。学院旨在培养科学和人文学科——埃奇沃思的父亲是学院的创始人之一——但汉密尔顿认为文学没有得到足够的重视。“全世界都知道，”他写信给她，“你不仅是一个文学爱好者，还是一个成功的文学追求者和强有力的推动者，在与文学相关的任何方面，你的意见一定是最有价值的。”埃奇沃思的作品今天可能已被人们遗忘——尽管其中两部作品在 2009 年被《卫报》列为“每个人都必须阅读的 1000 部小说”——但在 19 世纪初，她是爱尔兰最著名的小说家，不仅汉密尔顿崇拜她，沃尔特·斯科特和简·奥斯汀等人也崇拜她。她给了汉密尔顿一些中肯的建议——比如为征文比赛提供奖牌，将订阅费定在普通文学爱好者可以承受的水平——汉密尔顿在就职演讲中感激地引用了其中的大部分建议。不幸的是，他没有足够的勇气去接受她关于允许女性参加学院特别讨论会的建议。她长期以来一直在倡导女性受教育的权利和参与科学的权利，而汉密尔顿在回复中找的借口让她很不满意。尽管如此，他还是主持了她 1842 年的学院荣誉院士选举——她是第一位获得此殊荣的本地女性。此前已有三名外国女性获得这一殊荣——包括玛丽·萨默维尔和卡罗琳·赫歇尔（约翰·赫歇尔的姨妈，也是和她一样的天文学家）她们是英国女童军的一名成员（包括她的侄子和她的兄弟威廉）——但由于她们生活在国外，她们不太可能因为想要参加会议而惹恼男性。直到 1949 年，第一批正式女性成员才被接纳^。24

At the end of 1837, thirty-two-year-old Hamilton was elected president of the Royal Irish Academy, and one of the first things he did was seek advice from Maria Edgeworth, who had just turned seventy. The Academy was designed to foster science and the humanities—Edgeworth’s father had been a founder—but Hamilton felt that literature wasn’t getting enough of a look-in. “It is known to all the world,” he wrote to her, “that you are not only a lover of literature but a successful pursuer and powerful promoter of it, and that on any point connected therewith, your opinion must be most valuable.” Edgeworth’s works may be largely forgotten today—although in 2009 two of them made the Guardian’s list of “1000 novels everyone must read”—but in the early nineteenth century she was Ireland’s most famous novelist, and it wasn’t just Hamilton who admired her but the likes of Walter Scott and Jane Austen, too. She gave Hamilton some sound advice—such as offering medals for essay competitions, and setting subscription fees at an affordable rate for ordinary literaturelovers—and he gratefully drew on most of it in his inaugural presidential address. Unfortunately, he wasn’t brave enough to tackle her advice to allow women to attend special Academy discussion nights. She had long been promoting women’s right to education and their right to engage with science, and she wasn’t impressed with the excuses he made in his reply to her. Still, he would preside over her election as an honorary member of the Academy in 1842—the first local woman to be so honoured. Three foreign women had already achieved this distinction—including Mary Somerville and Caroline Herschel ( John Herschel’s aunt, and an astronomer like her nephew and her brother William)—but, living abroad, they were unlikely to ruffle male feathers by wanting to attend meetings. The first full female members would not be admitted until 1949.²⁴

令人失望的是，汉密尔顿在所谓的女性问题上并不像在数学上那样领先于时代，但他接下来的所作所为对于向量的故事至关重要。首先，虽然高斯和其他人已经看到复数可以表示为复平面上的一个点，但汉密尔顿将其视为“有向线段”——换句话说，一个箭头。25^韦塞尔也有类似的想法，但汉密尔顿和主流中的其他人都不知道。阿尔冈德也考虑过“有向线段”，但汉密尔顿的下一步才是向量的真正基础。如果你将两个“对”——阿尔冈德平面上的两个“箭头”（x，y）和（a，b）相乘——你会得到二维平面上的旋转，如图 3.8所示。所以复数乘法是与旋转的几何运算有关——就像乘以i一样。汉密尔顿想知道，是否可以通过“三元组”的乘法来表示三维空间中的旋转？

Disappointing as it is that Hamilton was not as ahead of his time on the so-called woman question as he was in mathematics, it is what he did next that is important for the story of vectors. First, while Gauss and others had seen that a complex number could be represented as a point in the complex plane, Hamilton saw it as a “directed line segment”—in other words, an arrow.²⁵ Wessel had something like this idea, too, but neither Hamilton nor anyone else in the mainstream knew about it. Argand also considered “directed line segments,” but it was Hamilton’s next step that would be truly foundational for vectors. If you multiply two “couples”—two “arrows” (x, y) and (a, b) in the Argand plane—you get a rotation in a twodimensional plane, as in figure 3.8. So complex number multiplication is linked to the geometrical operation of rotation—just as it is for multiplication by i. Was it possible, Hamilton wondered, to represent rotations in three-dimensional space via the multiplication of “triples”?

1841 年，他开始认真研究这个问题，当时他读到一篇论文，其中他的朋友德·摩根 (De Morgan) 总结说，用代数表示三维几何似乎根本不可能。虚数i在二维空间中发挥了旋转的魔力，但德·摩根认为，没有更多这样的符号可用——没有新的代数思想可以将几何从二维页面转移到整个空间。

He began working on the problem in earnest in 1841, when he read a paper in which his friend De Morgan concluded that it simply didn’t seem possible to represent three-dimensional geometry algebraically. The imaginary number i had performed its rotational magic in two dimensions, but De Morgan argued that there were no more such symbols available—no new algebraic ideas that could move geometry from the two-dimensional page to all of space.

这正是汉密尔顿所需要的挑战。

It was just the challenge Hamilton needed.

汉密尔顿多年来一直在思考他的三元组代数，到 1843 年秋天，甚至他的孩子们也开始关注他的传奇故事。尽管他们还很小，汉密尔顿还是试图向八岁的阿奇博尔德和九岁的威廉·埃德温解释他的基本思想，现在每天早上吃早餐时，他们都会问：“爸爸，你会乘三元组吗？”汉密尔顿每天早上都会悲伤地摇头说：“不，我只会加减。” ¹

Hamilton mulled over his algebra of triples for years, and by the fall of 1843 even his children were following the saga. Although they were very young, Hamilton had tried to explain his basic idea to eight-year-old Archibald and nine-year-old William Edwin, and now every morning at breakfast they would ask, “Well, Papa, can you multiply triplets?” And every morning Hamilton would shake his head sadly, saying, “No, I can only add and subtract them.”¹

汉密尔顿的几封信都动人地提到了他向“我的孩子们”解释数学的故事，就三元组而言，起初，答案似乎是小菜一碟——至少对汉密尔顿这样的神童来说是这样。毕竟，乘以一对复数实数（x，y）相当于在二维平面上旋转有向线OP = x + iy ，如图3.6 - 3.8所示——所以在三维空间中你肯定会有“三元组”（x，y，z），而你只需旋转OP = x + iy + jz。当然，你必须发明j，另一个像i一样的虚数，但这似乎很简单：如果你将阿尔冈平面扩展到三维，j在第三轴方向上，你会得到一个新的虚数，由于方向不同，它与i不同。如果你通过类比i来定义j的乘法——即在x - z平面上旋转——你会得到j ² = −1。（事实上，今天的工程师用j来表示$\sqrt{-1}$，因为他们用i来表示电流。）但是，这个新数字j与原来的i之间的代数差是什么？

Several of Hamilton’s letters have touching references to his explaining maths to “my boys,” and in the case of triples it had seemed, at first, as if the answer would be child’s play—for a prodigy such as Hamilton, at least. After all, multiplying by a complex “couple” of real numbers (x, y) was equivalent to rotating the directed line OP = x + iy in the two-dimensional plane, as in figures 3.6–3.8—so surely in 3-D space you’d have “triples” (x, y, z), and you’d just be rotating OP = x + iy + jz. You’d have to invent j, of course, another imaginary number like i, but that seemed straightforward enough: if you extended the Argand plane to three dimensions, with j in the direction of the third axis, you’d have a new imaginary number, different from i because of its direction. And if you defined multiplication by j via analogy with i—that is, as a rotation in the x-z plane—you’d have j² = −1. (In fact, today engineers use j to denote $\sqrt{-1}$, because they use i to denote electric current.) But then, what was the algebraic difference between this new number j and the original i?

汉密尔顿做的第一件事就是研究代数定律，德·摩根和皮科克已经开始将其形式化。这似乎是显而易见的。如果你将数字相加，如x + iy + jz和a + ib + jc，那么以何种顺序相加并不重要，而且你可以相当肯定结合律也会成立。（我在尾注中对这些定律进行了提醒^。2）

The first thing Hamilton did was to check in with the laws of algebra, which De Morgan and Peacock had begun to formalise. It seemed obvious that if you added numbers like x + iy + jz and a + ib + jc, it wouldn’t matter in which order you added them, and you could be pretty sure that the associative law would hold, too. (I’ve given a reminder of these laws in the endnote.²)

然后是代数“闭包”的概念，正如现代数学家所说的那样。如果将两个整数相加，会得到另一个整数；如果将两个实数相加，会得到另一个实数；如果将两个复数相加，会得到另一个复数。很明显，如果将两个“三元组”相加，闭包也成立。所以这一切都很简单。乘法会很棘手。

Then there’s the idea of algebraic “closure,” as modern mathematicians put it. If you add two integers, you get another integer; if you add two real numbers, you get another real number; and if you add two complex numbers, you get another complex number. It’s obvious that closure holds if you add two “triples,” too. So that was all nice and easy. It was multiplication that was going to be tricky.

汉密尔顿通过展开（ x + iy）（a + ib ）找到了乘以他的数对的规则。但是，如果逐项相乘（x + iy + jz）和（a + ib + jc），您会发现不仅需要i ² = j ² = −1，还必须弄清楚ij和ji的含义。从表面上看，这些额外的项表明三元组的乘法不是封闭的 - 对于纯数学家来说，这意味着三元组不形成一致的乘法代数。（如果您永远不知道乘积中会得到什么类型的数字，那么您就无法设计出适用于初始集合中每个可能数字的一致代数规则 - 如果没有规则，您就无法解方程。）

Hamilton had found the rules for multiplying his couples simply by expanding (x + iy)(a + ib). But if you multiply out (x + iy + jz) and (a + ib + jc), term by term, you can see that you’re not only going to need i² = j² = −1, you’re also going to have to figure out what ij and ji mean. On the face of it, these extra terms suggest that multiplication of triples is not closed— and to a pure mathematician, that means triples do not form a consistent multiplicative algebra. (If you never know what type of number you’ll get in a product, you can’t devise consistent algebraic rules that work for every possible number in your initial set—and if there are no rules, you can’t solve equations.)

按照良好的数学研究风格，汉密尔顿首先试图简化问题，假设ij = 0。这当然会使乘法封闭，但这有点牵强——在普通算术和代数中，不能将两个非零量相乘得到零。（如果一个或两个非零矩阵的行列式为零，则可以从两个非零矩阵的乘积中得到零，例如当矩阵方程AX = 0 时——但矩阵代数是在四元数之后出现的。）还有其他定律需要检查——汉密尔顿特别担心他所谓的“模数定律”，我在尾注中对此进行了解释。3^然而，为了让这个定律适用于三元组，汉密尔顿发现ij不能等于0。正如他告诉约翰·格雷夫斯——他的大学老友，他也一直在考虑扩展复平面，他的兄弟罗伯特后来写了一本对汉密尔顿充满同情的传记——他随后绝望地尝试了ij = − ji，尽管它与普通算术相反，就像ij = 0 一样。他甚至尝试了ij = j和ji = i。但是，尽管所有这些奇怪的可能性确实简化了乘法的某些方面，但他的新数字仍然不满足模数定律^。4

In good mathematical research style, Hamilton started out by trying to simplify matters, assuming ij = 0. That certainly would make multiplication closed, but it was a stretch—in ordinary arithmetic and algebra, you can’t multiply two nonzero quantities and get zero. (You can get zero from the product of two nonzero matrices if the determinant of one or both is zero, such as when you have the matrix equation AX = 0—but matrix algebra came after quaternions.) And there were other laws to check, too— Hamilton was worried especially about what he called the “law of moduli,” which I’ve explained in the endnote.³ To make this work for triples, however, Hamilton found that ij could not equal 0. As he told John Graves—his old friend from university, who had also been thinking about extending the complex plane, and whose brother Robert would later write a sympathetic biography of Hamilton—he then tried ij = −ji in desperation, even though it was as contrary to ordinary arithmetic as ij = 0. He even tried ij = j and ji = i. But while all these bizarre possibilities did simplify some aspects of multiplication, his new numbers still didn’t satisfy the law of moduli.⁴

汉密尔顿一定很想放弃。也许这真的是不可能的，就像他的朋友德·摩根所说的那样。

Hamilton must have been sorely tempted to give up. Perhaps it really was impossible, just as his friend De Morgan had said.

和汉密尔顿一样，德·摩根在还不到 22 岁时就担任了教授，当时他在英国第一所非宗派大学，即新成立的伦敦大学（不久后更名为伦敦大学学院）担任教授。两人出生时间相差不到一年，尽管德·摩根在婴儿时期就失去了一只眼睛。他曾因视力残疾而在学校受到欺负，但和欧拉一样，他没有让视力残疾阻碍自己成为一流的数学家。

Like Hamilton, De Morgan was not quite twenty-two when he’d first taken up a professorship—in his case, at England’s first nonsectarian university, the newly established London University, soon renamed University College, London. The two men were also born within a year of each other, although as a baby De Morgan had become blind in one eye. He’d been bullied at school for his ocular disability, but like Euler, he didn’t let it prevent him from becoming a first-rate mathematician.

德·摩根很幸运，他的妻子索菲亚·弗伦德是一位受过良好教育的社会活动家、辩论家和唯灵论者，与他志趣相投。他们都对自由充满热情，以至于他们采取了不同寻常的措施，在登记处结婚，以此来宣布他们的宗教独立。相比之下，汉密尔顿有着他的一位门徒所说的“深深的虔诚天性”，他是一个虔诚的英国国教徒，而他的妻子海伦据说特别虔诚^。6

De Morgan was lucky enough to find a kindred soul in his wife, the well-educated social activist, polemicist, and spiritualist Sophia Frend. They were both so passionately committed to freedom that they’d taken the unusual step of getting married in a registry office, as a way of declaring their religious independence. By contrast, Hamilton had what one of his disciples would call “a deeply reverential nature,” and he was a committed Anglican, while his wife Helen was reputedly particularly religious.⁶

汉密尔顿也是一个情绪化的人，没有海伦在身边，她就无法轻松地工作。但有时——尤其是当她怀孕或哺乳时——她在距离都柏林五英里的邓辛克天文台黑暗偏僻的场地上感到焦虑，自从被任命为爱尔兰皇家天文学家以来，汉密尔顿就住在那里。她开始离家出走，住在附近的亲戚家，汉密尔顿经常和她住在一起。但是，在 1841 年——他们的女儿海伦·伊丽莎出生一年左右后——她离开了丈夫和孩子，和她在英国的姐姐住在一起。她离开了大约一年，各种各样的谣言开始四处流传，关于这场所谓的灾难性的婚姻，以及汉密尔顿似乎开始酗酒。他失去了他的初恋凯瑟琳·迪斯尼，这并没有帮助，因为当时严格的浪漫礼仪导致了误解，简·奥斯汀在十年或二十年前的小说中就强调了这一点。 19 世纪 30 年代和 40 年代，随着禁酒运动的兴起，人们对社交饮酒的态度急剧收紧，这也于事无补。因此，每过十年，这些故事就变得越来越夸张，不公平地玷污了汉密尔顿在世时的名声，直到今天仍然如此^。7

Hamilton was also an emotional man and couldn’t work easily without Helen nearby. But sometimes—especially when she was pregnant or nursing—she felt anxious in the dark, remote grounds of Dunsink Observatory, five miles from Dublin, where Hamilton had lived since his appointment as Ireland’s royal astronomer. She began spending time away from home, staying with relatives nearby, and Hamilton often stayed with her there. But then, in 1841—a year or so after their daughter Helen Eliza was born—she left her husband and children to stay with her sister in England. She was away for about a year, and all sorts of rumours began to fly around, about the supposedly disastrous marriage and how Hamilton had apparently taken to drink. It didn’t help that he’d famously lost his first love, Catherine Disney, through the kind of misunderstandings engendered by the strict romantic etiquette of the day, which Jane Austen had highlighted in her novels just a decade or two earlier. It didn’t help, either, that attitudes to social drinking dramatically tightened in the 1830s and 1840s, as the temperance movement gathered pace. So, with each passing decade the stories became more and more exaggerated, unfairly tainting Hamilton’s reputation during his lifetime and continuing right up until today.⁷

虽然最近的研究表明，汉密尔顿夫妇的婚姻比传言中幸福得多，但德·摩根夫妇似乎过着幸福的家庭生活。索菲亚的父母是诗人拜伦夫人的朋友，拜伦夫人是这位诗人的离异妻子。1840 年至 1842 年的几年里，汉密尔顿夫妇正在应对海伦的焦虑和疾病，德·摩根则在辅导拜伦家的杰出女儿艾达·洛芙莱斯。（她的第一位数学导师是玛丽·萨默维尔，后来她搬到了气候更健康的意大利。）一年后，洛芙莱斯写了她对查尔斯·巴贝奇设计的先进计算机进行了数学开发——她的评论包括通常被认为是第一个计算机程序。它发表于汉密尔顿发现四元数的同一个月——事实证明，这是一个预言性的时间巧合，因为我们很快就会看到，正是计算机编程揭示了四元数的真正计算经济性^。8

While recent scholarship suggests that the Hamiltons had a much happier marriage than the rumours suggest, the De Morgans seem to have had a charmed domestic life. Sophia’s parents were friends of Lady Byron, the poet’s estranged wife, and for a couple of years between 1840 and 1842, while the Hamiltons were dealing with Helen’s anxiety and ill health, De Morgan was tutoring the Byrons’ remarkable daughter, Ada Lovelace. (Her first mathematical mentor had been Mary Somerville, who had since moved to Italy for the healthier climate.) A year later, Lovelace wrote up her mathematical development of Charles Babbage’s design for an advanced calculating machine—and her commentary included what is often considered the first computer program. It was published in the very same month that Hamilton discovered quaternions—a prophetic coincidence of timing, as it turns out, for we’ll see shortly that it is computer programming that has brought to light the true computational economy of quaternions.⁸

与此同时，汉密尔顿仍在为寻找四元数而苦苦挣扎，当德摩根声称三维复数代数不可能时，汉密尔顿将其视为一个严肃的挑战。毕竟，他和德摩根都在寻找理解代数的方法——给它一个逻辑基础，就像两千年前欧几里得时代以来几何学所拥有的那样。正是这种欧几里得的严谨性让牛顿从几何上证明了《自然哲学的数学原理》的定理，而不是通过更紧凑但在方法论上仍然不稳定的代数微积分。至于更基本的代数问题，我已经指出，正是由于缺乏对负数和虚数的具体解释，早期的数学家们才放弃了它们——但正如德摩根在这种情况下讽刺地指出的那样，“没有什么比拒绝所有可能带来麻烦的东西更能让心灵放松的了。” ⁹

Meantime, Hamilton was still struggling in his quest for quaternions, and when De Morgan claimed the impossibility of a three-dimensional complex algebra, Hamilton had taken it as a serious challenge. After all, both he and De Morgan were searching for ways to make sense of algebra—to give it a logical foundation, the way geometry had had since the time of Euclid two thousand years earlier. It was this Euclidean rigour that had led Newton to justify the theorems of Principia geometrically, rather than via more compact but still methodologically shaky algebraic calculus. As for more fundamental algebraic problems, I’ve already indicated that it was the lack of a concrete explanation of negative and imaginary numbers that had led earlier mathematicians to dismiss them—but as De Morgan wryly noted in this context, “Nothing could make a more easy pillow for the mind, than the rejection of all which could give any trouble.”⁹

汉密尔顿绝对不是那种轻轻松松就能找到安乐窝的人。所以他一直埋头苦干，先尝试这个，然后尝试那个，从未放弃这个想法：如果你能用实数 ( x, y ) 来表示普通二维复数的代数，那么你就应该能够将其扩展为 ( x, y, z )——甚至，正如他在 1841 年告诉德·摩根的那样，可以扩展到任意维数，例如a = ( a ₁ , a ₂ , … , _an )，其中a ₁ , a ₂ , … , an是实数。他很有_先见之明地说，单个符号a “表示一个（复杂的）思想”。他继续从他的数学时间哲学的角度阐述这一点，向德·摩根建议，这n 个数字——你可以根据问题选择n的任意值——表示事件的时间顺序，因此是一种“因果关系的概念”。尽管汉密尔顿最终不再迷恋时间，但他还是瞥见了一些重要的东西：一个符号可以编码许多信息，或者“一个复杂的想法”。正如我所提到的，这是向量和张量的力量的一部分，它也有助于解释四元数的计算经济性^。10

Hamilton was definitely not one to use an easy pillow. So he kept plodding away, trying first this and then that, and never giving up on the idea that if you could write the algebra of ordinary, two-dimensional complex numbers in terms of real numbers (x, y), then you should be able to extend it to (x, y, z)—and even, as he’d told De Morgan in 1841, to any number of dimensions, as in a = (a₁, a₂, … , a_n), where the a₁, a₂, … , a_n are real numbers. The single symbol a, he said presciently, is “indicative of one (complex) thought.” He went on to elaborate this in terms of his mathematical philosophy of time, suggesting to De Morgan that these n numbers—where you can choose any value of n, depending on the problem—represent a chronological ordering of events, and therefore a “notion of cause and effect.” But while Hamilton would ultimately move on from his fascination with time, he had glimpsed something important here: the idea that a single symbol can encode many pieces of information, or “one complex thought.” As I’ve mentioned, this is part of the power of vectors and tensors, and it will also help explain the computational economy of quaternions.¹⁰

汉密尔顿在探索过程中付出了巨大的努力，以至于他的书房里堆满了文件，散落在桌子上，堆在地板上，他经常不吃饭，靠带到房间里的零食维持生活。但后来有一天，他的坚持得到了回报，他突然有了一个惊人的想法。正如他向约翰·格雷夫斯解释的那样，“我突然想到，在某种意义上，我们必须承认空间存在第四维。”他的意思是，如果他引入第三个虚数k——这立即暗示了一个新的“4-D”复数a + ib + jc + kd的存在——并且如果他将k定义为ij，那么三元组的模数定律最终就会起作用。你可以在下面的尾注中看到原因，但正是这个奇妙的洞察力打动了他，就像“电路闭合，火花闪现”，1843 年秋天那天，他和海伦在都柏林的布鲁姆桥上散步^。11

Hamilton worked so hard in his quest that his study was a mess of papers, scattered over his desk and piled high on the floor, and he often skipped meals, surviving on snacks brought to his room. But then one day his persistence paid off, and suddenly he had a startling idea. As he explained to John Graves, “there dawned on me the notion that we must admit, in some sense, a fourth dimension of space.” What he meant was that if he introduced a third imaginary number, k—which immediately suggests the existence of a new “4-D” complex number a + ib + jc + kd—and if he defined k to be ij, then the law of moduli for triples would work at last. You can see why in the following endnote, but it was this marvelous insight that struck him, as if “an electric circuit closed and a spark flashed forth,” as he and Helen were walking by Broome Bridge on that autumn Dublin day in 1843.¹¹

在命名他的新 4-D 数字时，我不知道他为什么不在“对”和“三元组”或“三重奏”后面加上“四元组”或“四重奏”——但“四元数”听起来确实很顺耳，而汉密尔顿对诗歌很有鉴赏力。正如德摩根后来回忆的那样，“汉密尔顿自己说过，‘我靠数学生活，但我是一个诗人。’这样的格言可能会让读者感到惊讶，但他们应该记住，数学发明的推动力不是推理，而是想象力。”然而，汉密尔顿的朋友华兹华斯认为，汉密尔顿的想象力更适合数学而不是诗歌。他确实非常有创造力，开创了矢量分析，而且根据量子先驱埃尔温·薛定谔的说法，他还开创了量子力学的核心数学^。12

When it came to naming his new 4-D numbers, I’m not sure why he didn’t follow his “couples” and “triples” or “triplets” with “quadruples” or “quadruplets”—but “quaternion” does have a ring to it, and Hamilton had an ear for poetry. As De Morgan would later recall, “Hamilton himself said, ‘I live by mathematics, but I am a poet.’ Such an aphorism may surprise our readers, but they should remember that the moving power of mathematical invention is not reasoning, but imagination.” Hamilton’s friend Wordsworth, however, felt that Hamilton’s imagination was better directed toward mathematics than poetry. And he was indeed wonderfully inventive, pioneering both vector analysis and, according to no less an authority than quantum pioneer Erwin Schrödinger, the maths at the heart of quantum mechanics.¹²

汉密尔顿在看到这一惊人发现后非常兴奋，当场就在石桥上刻下了一条简单的线，

Hamilton had been so excited after his electric insight that right there and then he’d carved into the stone bridge a single, simple line,

i2 = j2 = ^k2 = ^ijk = −1 。^​

i² = j² = k² = ijk = −1.

它包含了他所说的“用几何计算”所需的一切——包括计算三维空间中的旋转。为此简单方程式是一种简洁的方式来写下他的三个虚数之间的必要代数关系：

It contained all that he needed to “calculate with geometry,” as he put it—including calculating rotations in three-dimensional space. For this line of simple equations was a compact way of writing down the necessary algebraic relationships between his three imaginary numbers:

ij = k = − ji, jk = i = − kj, ki = j = − ik。

ij = k = −ji, jk = i = −kj, ki = j = −ik.

（如果你用代数方法玩弄它，你就会从涂鸦中看到这一点：例如，k ² = ijk ⇔ k = ij。）¹³使用这些定义，如果你有耐心进行 4-D 乘法，例如

(You can see how this follows from the graffiti if you play with it algebraically: for instance, k² = ijk ⇔ k = ij.)¹³ Using these definitions, if you had the patience to carry out 4-D multiplications such as

( a + ib + jc + kd )( w + ix + jy + kz ),

(a + ib + jc + kd)(w + ix + jy + kz),

你会发现四元数不仅满足模数定律，还满足除交换律之外的所有普通实数和复数乘法定律。换句话说，

you’d see that quaternions satisfy not only the law of moduli, but all the laws of ordinary real and complex multiplication—except the commutative law. In other words,

( a + ib + jc + kd )( w + ix + jy + kz )≠( w + ix + jy + kz )( a + ib + jc + kd )。

(a + ib + jc + kd)(w + ix + jy + kz) ≠ (w + ix + jy + kz)(a + ib + jc + kd).

最后一个结论具有启示性，你无需进行繁琐的乘法运算就能猜到，因为虚数乘积本身（ij等）不具有交换性。对于汉密尔顿来说，这是一个不情愿的举动，他多年来一直试图将三元组和四元组纳入通常的算术法则，因此他值得称赞，因为他有足够的勇气迈出这一步，并认识到他已经发现了一种全新的代数。

It’s this last conclusion that was revelatory, and you can guess it without doing all the tedious multiplications, because the imaginary products themselves—ij and so on—are not commutative. This was a reluctant step for Hamilton, who had been working for years trying to fit first triples and then quaternions into the usual laws of arithmetic, so he deserves kudos for being brave enough to take it—and to recognise that he’d hit upon a brand-new kind of algebra.

汉密尔顿的虚向量ix + jy + kz看起来很像你可能认识的向量x i + y j + z k，其中i、j、k以粗体显示，因为它们不再是基本虚数——而是实数“单位向量”，其大小或长度为“1”（即 1）。所以你已经可以看到当今向量的起源，尽管在我们了解这是如何发生的之前，故事还有很长的路要走。

Hamilton’s imaginary vector ix + jy + kz looks very like the vector you might recognise as xi + yj + zk, where the i, j, k are in boldface type because they are no longer fundamental imaginary numbers—rather, they’re real “unit vectors,” vectors whose magnitude, or length, is “unity” (which just means 1). So already you can see the genesis of today’s vectors, although there’s still a way to go in the story before we see how this happened.

四元数符号P编码了四部分信息——标量w和矢量的三个分量x、y、z——因此你可以将P视为代表“一个复杂的思想”，就像汉密尔顿在 1841 年写给德·摩根的焦急信中所预见的那样。

The quaternion symbol P encodes four pieces of information—the scalar w and the vector’s three components x, y, z—so you can think of P as representing “one complex thought,” just as Hamilton had foreseen in his anxious letter to De Morgan back in 1841.

如果我们从未来中抄袭粗体字，以区分汉密尔顿的矢量和他的标量，我们可以将四元数P写成，其虚数“基”为i, j, k，如下

If we crib the bold type from the future to distinguish Hamilton’s vector from his scalar, we can write the quaternion P, with its imaginarynumber “basis” i, j, k, as

P = w + p，其中p = ix + jy + kz。

P = w + p, where p = ix + jy + kz.

（汉密尔顿有时把虚数放在实数分量之后，就像我们处理向量一样，因此ix, jy, kz中虚数和实数的顺序无关紧要 - 如果你愿意，可以将其视为xi, yj, zk。）“基”本质上只是一组独立的单位量，每个轴一个。更具体地说，这些基量被认为跨越进行计算的向量空间，但这些概念尚未得到正确的表述；这里的重点是，如果你从单位虚数基（ i，j，k）更改为单位向量基（i，j，k ） ，也可以将信息写在向量p = ix + jy + kz中，并写为p = x i + y j + z k。这是因为关键在于在每种情况下，分量都有相同的数值（x，y，z），因为无论哪种方式你都使用笛卡尔坐标系。它是你用矢量进行计算时所需要的分量。

(Hamilton sometimes put the imaginary numbers after the real number components as we do with vectors, so the order of the imaginary and real numbers in ix, jy, kz doesn’t matter—you can think of it as xi, yj, zk if you prefer.) A “basis” is essentially just a set of independent unit quantities, one along each axis. More technically, these basis quantities are said to span the vector space in which the calculations take place, but these notions weren’t yet properly formulated; the point here is that you can write the information in the vector p = ix + jy + kz just as well if you change from the unit imaginary basis (i, j, k) to the unit vector basis (i, j, k), and write p = xi + yj + zk. That’s because the key thing is that in each case the components have the same numerical values, (x, y, z)—because either way you’re using a Cartesian coordinate system. And it is the components you need when you calculate with vectors.

换句话说，你可以将向量视为一种用于存储数据的设备（以组件的值形式存储），而基础则是设备的硬件。不同的设备具有不同的设计（在本例中是实数基础i、j、k或虚数基础i、j、k），但它们都存储相同的信息。不过，这只是事后的看法：从汉密尔顿向量到现代向量的令人惊讶的激烈转变是第 8 章的故事，但我在这里提到这一点是为了让熟悉现代向量的读者了解。

In other words, you can think of a vector as a device for storing data—in the values of the components—while the basis is the device’s hardware. Different devices have different designs—in this case a real basis, i, j, k or an imaginary one, i, j, k—but they both store the same information. This is hindsight, though: the surprisingly acrimonious transition from Hamilton’s vectors to modern ones is a story for chapter 8, but I mention it here for readers familiar with modern vectors.

如果你熟悉向量，那么汉密尔顿的四元数可能看起来很奇怪：又是一个苹果和橘子的例子，因为在本科向量分析中，我们不会把向量加到标量上。但在这个阶段，我们需要把向量简单地看作一个四维复数的虚部，就像汉密尔顿所做的那样——一个遵循他在布鲁姆桥上雕刻的虚数规则的代数表达式。

If you are familiar with vectors, Hamilton’s quaternion might seem strange: another case of apples and oranges, because adding a vector to a scalar is not something we do in undergrad vector analysis. But at this stage of the story, we need to think of a vector simply as the imaginary part of a 4-D complex number, as Hamilton did—an algebraic expression that follows the rules of the imaginary numbers he carved on Broome Bridge.

考虑到这一点，现在取第二个四元数，

With this in mind, now take a second quaternion,

Q = a + ib + jc + kd（或a + bi + cj + dk）= a + q。

Q = a + ib + jc + kd (or a + bi + cj + dk) = a + q.

汉密尔顿发现，当你将P和Q的乘积逐个分量相乘，并将所有得到的项分组为标量或矢量时，你会得到两种新的乘法，这里用现代的点和叉表示：

Hamilton discovered that when you multiply out the product of P and Q, component by component, and when you group all the resulting terms as scalars or vectors, you get two new kinds of multiplication, indicated here by the modern dot and cross:

PQ = wa − p ∙ q + wq + ap + p × q。​​

PQ = wa − p ∙ q + wq + ap + p × q.

这是一个非常紧凑的表达式，因为它实际上包含二十二次乘法和十一次加法或减法。这种非凡的经济性是四元数和整个矢量（用特殊符号表示，比如这里的粗体，而不是列出它们的分量）威力的一个例子。除了减号之外，计算上汉密尔顿的“标量”和“矢量”积——分别是p ∙ q和p × q ——与现代的完全相同；因此，为了了解四元数（和整个矢量）乘法的紧凑性，我在尾注中给出了这些乘积的现代分量形式，尽管你可能在数学课上熟悉它们。14^但你可能不知道，今天经常用于矢量的粗体字是由特立独行的英国人奥利弗·亥维赛 (Oliver Heaviside) 发明的。稍后我们会与彬彬有礼的美国人约西亚·威拉德·吉布斯 (Josiah Willard Gibbs) 一起正式会见他，他为我们提供了表示标量积的点 (现在也称为“点积”) 和表示矢量 (或叉) 积的叉。

This is an amazingly compact expression, given that it actually contains twenty-two multiplications and eleven additions or subtractions. This extraordinary economy is one example of the power of quaternions and whole vectors (the ones represented by special notation such as the boldface type here, rather than by listing their components). Aside from the minus sign, computationally Hamilton’s “scalar” and “vector” products— the p ∙ q and p × q, respectively—are just the same as the modern ones; so to see the compactness of quaternion (and whole-vector) multiplication, I’ve given the modern component forms of these products in the endnote, although you’re likely familiar with them from maths classes.¹⁴ But you may not know that the bold type often used today for vectors is due to the idiosyncratic Englishman Oliver Heaviside. We’ll meet him properly later, along with the urbane American Josiah Willard Gibbs, who gave us the dot for scalar products (also now called “dot products”) and the cross for the vector (or cross) product.

汉密尔顿本人并没有将这两个新乘积指定为矢量的乘积；相反，他把它们称为四元数乘积的“标量部分”和“矢量部分” 。然而，对于他的两个四元数P和Q，汉密尔顿将标量w和a设为零，并像我们今天一样将矢量分量相乘——只是他没有使用粗体字母和点或叉，而是将“标量部分”（我们的标量积）写为S.PQ ，将“矢量部分”（我们的矢量积）写为V.PQ。标量积是可交换的，但矢量积不是，这在称为“右手定则”的助记符中有所体现（图 4.1 ）。这个规则的意思是，当你沿着从矢量p到矢量q 的方向弯曲手指时，你的拇指指向乘积矢量p × q的方向。但是如果你反转乘法的顺序，你的拇指就会指向相反的方向。这意味着p × q = − q × p。由于向量积是非交换的，所以整个四元数积PQ也是非交换的。

Hamilton himself hadn’t designated these two new products as products of vectors; rather, he called them the “scalar part” and the “vector part” of the quaternion product. Nevertheless, for his two quaternions P and Q, Hamilton set the scalars w and a to zero and multiplied out the vector components just as we do today—except that instead of using boldface letters and a dot or a cross he wrote S. PQ for the “scalar part” (our scalar product), and V. PQ for the “vector part” (our vector product). The scalar product is commutative, but the vector product is not, and this is encoded in the mnemonic device called the “right-hand rule” (fig. 4.1). What this rule means is that when you curl your fingers in the direction from vector p to vector q, your thumb points in the direction of the product vector p × q. But if you reverse the order of the multiplication, your thumb points in the opposite direction. Which means that p × q = −q × p. And because vector products are noncommutative, so is the whole quaternion product PQ.

正如您在图表（以及上一个尾注）中看到的，向量或叉积的关键在于它们将两个向量相乘得到另一个向量；标量或点积将两个向量相乘得到一个标量。（这就是为什么有一个数学笑话问，“当你把登山者和蚊子杂交时，你会得到什么？”答案是“什么也没有：你不能将标量与向量杂交”——因为叉积只能应用于两个向量，而不能应用于标量和向量。这也突出了“媒介”的生物学定义，即将病原体从一种动物或植物“携带”到另一种动物或植物的生物。）

As you can see in the diagram (and in the previous endnote), the key thing about vector or cross products is that they multiply two vectors to get another vector; scalar or dot products multiply two vectors to get a scalar. (Which is why there’s a maths joke that asks, “What do you get when you cross a mountain-climber with a mosquito?” The answer is, “Nothing: you can’t cross a scaler with a vector”—because the cross product can be applied only to two vectors, not to a scalar and a vector. This also highlights the biological definition of “vector,” meaning an organism that “carries” pathogens from one animal or plant to another.)

这种非交换向量乘法在抽象代数领域是一件大事，用汉密尔顿的话来说，这是一个“令人好奇、几乎疯狂”的发现。因为这是四千年有记载的数学史上第一次有人发现一个自洽的代数系统，其中XY 不等于YX。这意味着有一个全新的代数世界可供探索，谁知道有多少潜在的应用。这就是为什么汉密尔顿经常被描述为“代数的解放者”。

This noncommutative vector multiplication was a Very Big Deal in the world of abstract algebra, a “curious, almost wild” revelation, as Hamilton put it. For it was the first time in four thousand years of recorded mathematical history that someone discovered a self-consistent algebraic system where XY didn’t equal YX. And this meant there was a whole new algebraic world to explore, with who knew how many potential applications. That’s why Hamilton is often described as “the liberator of algebra.”

例如，如今非交换代数是量子力学和广义相对论的基础。在量子力学中，海森堡著名的不确定性原理指出，你不能同时精确测量量子粒子的位置X和动量P。但你可以先精确测量其中一个属性，比如位置，然后再测量另一个。问题是，当你进行第二次测量时，你第一次进行的位置测量将不再那么准确，因为亚原子粒子总是在移动或振动——所以如果你再次测量，你会得到不同的答案。这是非交换性在起作用，它有一个特殊的方程（海森堡原理由此得出，X 和 P 实际上是矩阵）：

Today, for example, noncommutative algebras underpin quantum mechanics and general relativity. In quantum mechanics, Heisenberg’s famous uncertainty principle says that you cannot precisely measure a quantum particle’s position X and momentum P at the same time. But you can precisely measure first one of these attributes, say the position, and then the other. The trouble is that by the time you’ve made this second measurement, the position measurement you made first will no longer be quite so accurate, because subatomic particles are always moving or vibrating—so you’ll get a different answer if you measure it again. This is noncommutativity in action, and it has a special equation (from which Heisenberg’s principle follows, and X and P are, in fact, matrices):

$十磷-磷十=我\frac{\mathrm{时长}}{2\text{π}}。$

$XP-PX=i\frac{h}{2\text{π}}.$

h是普朗克常数，它代表着测量的基本极限，而且它非常小：以千克米² /秒（或焦耳秒）为单位，为 6.6260693 × 10 ⁻³⁴ 。无论它有多小， h都不为零，这就是关键所在——因为如果测量可观测位置和动量的顺序无关紧要，那么就会有XP − PX = 0。当然，这等同于XP = PX，而这正是交换律。

The h is Planck’s constant; it represents the fundamental limit to measurement, and it is tiny: 6.6260693 × 10⁻³⁴ in units of kg m²/second (or joule-seconds). Tiny or not, h is not zero, and that’s the point here—for if it didn’t matter in which order you measured the observable position and momentum, you’d have XP − PX = 0. This, of course, is the same as XP = PX, which is none other than the commutative law.

在广义相对论中，非交换性给出了时空曲率的度量，因此当我们讨论张量时，我们会重新讨论这一点。同时，您可能已经想到，除了向量之外，普通矩阵也提供了一个更熟悉的非交换乘法示例。

In general relativity, noncommutativity gives a measure of the curvature of space-time, so we’ll revisit this when we come to tensors. Meantime, you might already be thinking that along with vectors, ordinary matrices offer a more familiar example of noncommutative multiplication.

凯莱比汉密尔顿小 15 岁，生命中的前 8 年是在俄罗斯度过的（当时他的商人父亲住在圣彼得堡）。他以全班第一的成绩毕业于剑桥大学三一学院，并以临时奖学金在那里教了几年书。要成为正式的奖学金获得者，必须宣誓效忠圣公会，因此，尽管他是一个虔诚的圣公会教徒，但接受圣职还是太过分了，于是他在 1846 年开始接受律师培训。他几乎和欧拉一样多产，在担任房地产律师的 14 年里，他利用业余时间写了近 300 篇数学论文，之后又写了数百篇。他最早的一篇论文是在离开剑桥大学攻读法律之前写的，四元数——凯莱是第一个公开接受汉密尔顿思想的人。他于 1845 年发表了这篇论文，而汉密尔顿于 1843 年 11 月 13 日在爱尔兰皇家学院宣读的一篇论文中首次宣布了他的发现，不到一年。1848 年，汉密尔顿在都柏林大学就四元数做了四次系列讲座，凯莱当时也在场^。15

Cayley, who was fifteen years younger than Hamilton, had spent the first eight years of his life in Russia (when his merchant father was based in St. Petersburg). He graduated top of his class at Trinity College, Cambridge, and taught there for several years on a temporary fellowship. A full fellowship required taking Anglican vows, so although he was a committed Anglican, taking Holy Orders was a step too far, and in 1846 he began training as a lawyer. He was almost as prolific as Euler, writing nearly three hundred mathematical papers in his spare time during the fourteen years he worked as a property lawyer as well as hundreds more afterward. One of his earliest papers, written before he left Cambridge for the law, was on quaternions—Cayley was the very first to publicly take up Hamilton’s idea. He published this paper in 1845, barely a year after Hamilton had first announced his discovery in a paper he read to the Royal Irish Academy on November 13, 1843. And when Hamilton gave a series of four lectures on quaternions at Dublin University in 1848, Cayley was in the audience.¹⁵

和所有事物一样，凯莱对矩阵的研究并非凭空而来，高斯和名气不大的费迪南德·爱森斯坦等数学家的研究都为其提供了先例。爱森斯坦在遇到汉密尔顿后，受到启发开始研究数学，但不幸在 29 岁时因肺结核英年早逝。凯莱自己对行列式的研究也有先例——行列式是在矩阵出现之前开发的，尽管今天我们将其作为矩阵的性质来了解。即使在古代，中国数学家也从他们提出和求解联立线性方程的方式中瞥见了矩阵的概念，他们使用数组和一种现在称为高斯消元法的形式。 （“线性”方程只有x、y等，就像直线方程一样——它们不包括未知数的乘积或幂，比如抛物线或圆的“二次”方程中的x2 ^。除了描述直线的能力之外，线性方程还有许多有用的应用，比如优化——例如优化机器学习模型的准确性或优化业务成本。它们在描述如何以特定方式从一个坐标系转换到另一个坐标系时也至关重要——比如当你想物理旋转或平移笛卡尔系统或其中的向量时，正如我们稍后将在图 4.2中看到的那样。随着故事的进展，这种“线性变换方程”将发挥越来越重要的作用。）

Like everything, Cayley’s work on matrices didn’t arise in a vacuum, and there had been antecedents in the work of mathematicians such as Gauss and the lesser-known Ferdinand Eisenstein, who had been inspired to take up maths after he met Hamilton, but who died tragically early of tuberculosis when he was just twenty-nine. There were also antecedents in Cayley’s own work on determinants—which were developed before matrices, although today we learn about them as properties of matrices. Even in the ancient world, Chinese mathematicians had glimpsed the idea of matrices in the way they set out and solved simultaneous linear equations, using arrays and a form of what is now known as Gaussian elimination. (“Linear” equations have only x, y and so on, such as you get in the equations for straight lines—they don’t include products or powers of the unknowns, such as the x² in the “quadratic” equation for a parabola, say, or a circle. Besides their ability to describe straight lines, linear equations have many useful applications, such as optimisation—optimising the accuracy of a machine learning model, for example, or optimising business costs. They are also crucial in describing how to change from one coordinate system to another in particular ways—such as when you want to physically rotate or translate your Cartesian system, or the vectors in it, as we’ll see later in figure 4.2. Such “linear transformation equations” will have an increasingly important role to play as this story progresses.)

古代中国的“高斯消元法”在两千年前的《九章算术》中得以保存，它远远领先于时代。中世纪印度数学家也做了类似的事情——公元七世纪，婆罗门笈多甚至使用数组来求解二次方程。所有这些早期方法都是纯算法的——非常适合计算机！因此，它们是专门计算的——对每个问题都有一个隐含的“先这样做，然后那样做”的计划——而不是像符号矩阵代数那样的代数和通用方法（或者实际上像牛顿和莱布尼茨的符号微积分算法）。因为高斯消元法不使用矩阵乘法或加法。16^凯莱使用的和《九章论》那位不知名的作者偶然发现的联立方程的方便表示法相同——但是因为他受益于符号代数，这最终使他走向了矩阵理论。不过，一开始，他只是想像他的前辈一样解方程——只不过他研究的是他的朋友詹姆斯·西尔维斯特很快就称之为“不变量”的东西。17^稍后我们会更多地讨论不变量，因为它们是张量概念的基础，在相对论中尤其重要。它们还使用我刚才提到的线性变换方程的概念。

The ancient Chinese “Gaussian elimination” method survives in the two-thousand-year-old Jiuzhang Suanshu, or Nine Chapters on the Mathematical Art, and it was way ahead of its time. Medieval Indian mathematicians did something similar—and in the seventh century Brahmagupta even used arrays to solve quadratic equations. All these early approaches were purely algorithmic—ideal for a computer! So, they were specifically computational—with an implied “do this then that” plan for each problem—rather than algebraic and general like symbolic matrix algebra (or, indeed, like Newton’s and Leibniz’s symbolic calculus algorithms). For Gaussian elimination doesn’t use matrix multiplication or addition.¹⁶ Cayley was using the same kind of convenient representation of simultaneous equations that the unknown author of Nine Chapters had hit upon— but because he had the benefit of symbolic algebra, this eventually led him to the theory of matrices. To begin with, though, he was simply trying to solve equations, just as his ancient predecessors had done—except that he was studying what his friend James Sylvester would soon dub “invariants.”¹⁷ More on invariants later, for they are fundamental to the idea of a tensor, and they’re especially important in relativity theory. They also use the idea of linear transformation equations that I just mentioned.

19 世纪 40 年代初，凯莱开始与乔治·布尔通信，讨论变换和不变量的问题。今天，我们因布尔发展了计算机的基础“布尔”代数逻辑而想到布尔，但布尔的出现稍晚一些，当时汉密尔顿的四元数已经开启了全新的符号代数的可能性。这是四元数的发现不仅对它本身很重要，而且有助于激发一系列新的代数发现，从矩阵到符号逻辑。无论如何，凯莱当时还是剑桥大学一位年轻的数学教授，而自学成才的布尔在林肯管理着一所学校，尽管他很快就获得了科克的教授职位。两人就他们的研究交换了信件，凯莱开始希望他们能亲自见面——但他遗憾地发现，这两个城镇之间还没有铁路连接^。18

In the early 1840s, Cayley began corresponding on the subject of transformations and invariants with George Boole. Today we think of Boole for his development of the “Boolean” algebraic logic that underpins computers, but that came a little later, after Hamilton’s quaternions had opened up the possibility of completely new symbolic algebras. It’s another example of the way the discovery of quaternions was important not only for itself, but also for helping to inspire a whole range of new algebraic discoveries, from matrices to symbolic logic. Anyway, Cayley was still a young professor of mathematics at Cambridge, and the self-taught Boole was running a school in Lincoln, although he would soon gain a professorship in Cork. The two men exchanged letters on their research, and Cayley began to wish they could meet in person—but, he lamented, there was not yet a rail link between the two towns.¹⁸

凯莱的哀叹是对当时技术的一个有趣评论——提醒人们，第一条使用蒸汽机车运载乘客和货物的公共铁路才建成不到十年，而且还在扩建。然而，它们已经为纺织品等制成品创造了新的市场，而这反过来又推动了新的科学和技术创新。工业革命正在顺利进行——蒸汽机的发明就是其中的一部分——铁路项目本身也引发了新科学发现的新应用，例如使用全新的电磁电报发明来协调列车交通。电磁学的存在仅在二十年前才被发现，因此该应用强调了一方面经济学和企业家精神与另一方面科学、数学和技术之间的古老双向联系。

Cayley’s lament is an interesting comment on the technology of the times—a reminder that the first public railways using steam locomotives to carry passengers as well as goods had been built barely a decade earlier, and they were still being expanded. Yet already they were creating new markets for manufactured products such as textiles, which in turn would drive new scientific and technological innovations. The Industrial Revolution was well underway—the invention of the steam engine had been part of it—and the railway project itself was sparking new applications of new scientific discoveries, such as the use of the brand-new invention of electromagnetic telegraphy to coordinate train traffic. The very existence of electromagnetism had only been discovered twenty years earlier, so this application highlights the ancient, two-way connection between economics and entrepreneurship on the one hand and science, mathematics, and technology on the other.

不幸的是，这种联系常常忽视技术对环境的影响。当时，至少还有一个借口，如果无知也算借口的话：又过了 10 年，美国气候学家先驱尤妮丝·牛顿·富特（Eunice Newton Foote，艾萨克爵士的远亲）发表了关于大气二氧化碳加热效应的里程碑式论文——这项令人兴奋的 19 世纪新技术大肆宣扬这一点。富特和其他早期气候科学家并没有过多考虑未来的气候变化；他们关注的是地球过去的气候是什么样的，以及如何推断出现在的温度。19^然而，有些人对这种“进步”感到不安——最著名的例子可能是美国作家亨利·戴维·梭罗，1845 年，他隐居在马萨诸塞州康科德市瓦尔登湖畔的森林中，以便更贴近大自然。与他的诗人朋友拉尔夫·沃尔多·爱默生一样，他受到超验主义、新英格兰哲学和文学回归自然运动的启发——与汉密尔顿交往的英国浪漫主义诗人有某种大致的相似之处。

Unfortunately, it’s a nexus that too often ignores the environmental impact of technology. Back then, at least there was an excuse, if ignorance counts as an excuse: it would be another decade before the pioneering American climate scientist Eunice Newton Foote, a distant relative of Sir Isaac, published her landmark paper on the heating effect of atmospheric carbon dioxide—which much of this exciting new nineteenth-century technology was spouting profusely. Not that Foote and other early climate scientists were thinking much about future climate change; they were focussed on what Earth’s climate was like in the past, and how to deduce its present temperature.¹⁹ Some, however, were uneasy about this kind of “progress”—perhaps most famously the American writer Henry David Thoreau, who in 1845 withdrew to the woods by Walden Pond in Concord, Massachusetts, in order to live closer to nature. Like his poet friend Ralph Waldo Emerson, he was inspired by transcendentalism, New England’s philosophical and literary back-to-nature movement—a kind of loose parallel to the English Romantic poets with whom Hamilton associated.

不过，凯莱并没有考虑工业与科学的联系——他从事数学是因为热爱它。因此，他很快意识到矩阵不仅仅是表示方程的便捷方法——它们本身就是代数结构，有自己的代数规则，这并不奇怪。

Cayley wasn’t thinking about the industrial nexus with science, though—he was engaged in mathematics for the love of it. So it’s not surprising that he soon realised matrices were more than just convenient methods of representing equations—they were algebraic structures in their own right, with their own algebraic rules.

例如，使用代数符号，你可以写出联立线性方程

For example, using algebraic symbolism you can write the simultaneous linear equations

2x + y =

7x − 3y = 1

2x + y = 7

x − 3y = 1

矩阵形式如下：

in matrix form like this:

$\left(\begin{array}{cc}2& 1\\ 1& -3\end{array}\right)\left(\begin{array}{c}十\\ 是\end{array}\right)=\left(\begin{array}{c}7\\ 1\end{array}\right)$

$\left(\begin{array}{cc}2& 1\\ 1& -3\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}7\\ 1\end{array}\right)$

（顺便一提，$\left(\begin{array}{l}十\\ 是\end{array}\right)$和$\left(\begin{array}{l}7\\ 1\end{array}\right)$是“列向量”的例子——其中信息写成列而不是我到目前为止一直使用的行，例如（x，y）。无论哪种方式，信息都是相同的。）这样写这两个方程式需要“逐列”矩阵乘法的想法，但凯莱最初是在尝试寻找一种简洁、省力的方式来表示多个线性变换时，想到了将任何两个兼容矩阵相乘——而不仅仅是一个矩阵和一个向量。例如，如果要旋转坐标系然后平移轴，则需要变换方程来从x - y坐标旋转到x′ - y′坐标（正如我们很快就会在图 4.2中看到的那样）。然后，您需要从x′ - y′变换到x″ - y″才能平移轴。事实证明，将这些变换组合起来得到x - y （以x″ - y″表示）时，最终矩阵中的元素给出了逐行定义。而这个定义又是乘法不总是可交换的另一种情况。

(By the way, $\left(\begin{array}{l}x\\ y\end{array}\right)$ and $\left(\begin{array}{l}7\\ 1\end{array}\right)$ are examples of “column vectors”—where the information is written as a column rather than the rows I’ve been using so far, such as (x, y). It’s the same information either way.) Writing these two equations like this takes the idea of “row-by-column” matrix multiplication for granted, but Cayley initially got the idea for multiplying any two compatible matrices—not just a matrix and a vector—when he was trying to find a neat, labour saving way of representing multiple linear transformations. For example, if you want to rotate a coordinate frame and then translate the axes, you’ll need transformation equations to rotate from x-y coordinates to x′-y′ coordinates, say (as we’ll see shortly, in fig. 4.2). Then you’ll need to transform from x′-y′ to x″-y″ in order to translate the axes. It turns out that on combining these transformations to get x-y in terms of x″-y″, the entries in the final matrix give the row-by-column definition. And this definition turns out to be another case where multiplication is not always commutative.

换句话说，一组传统线性方程中的信息可以以矩阵形式表示，就像古代中国数学家所看到的那样，但在凯莱手中，这导致了这样的想法：矩阵本身可以相乘，而不管它们代表什么。因此，他认为它们也可以相加：要添加两个矩阵（大小相同），只需在相应的位置添加两个数字，如下所示：

In other words, the information in a set of conventional linear equations can be represented in matrix form, as the ancient Chinese mathematicians had seen, but in Cayley’s hands this led to the idea that matrices themselves could be multiplied, regardless of what they represented. So, he figured that they could be added, too: to add two matrices (of the same size), you just add the two numbers at the corresponding places, like this:

$\left(\begin{array}{cc}3& -1\\ 2& 0\end{array}\right)+\left(\begin{array}{cc}1& 5\\ -2& 6\end{array}\right)=\left(\begin{array}{cc}4& 4\\ 0& 6\end{array}\right)$

$\left(\begin{array}{cc}3& -1\\ 2& 0\end{array}\right)+\left(\begin{array}{cc}1& 5\\ -2& 6\end{array}\right)=\left(\begin{array}{cc}4& 4\\ 0& 6\end{array}\right)$

凯莱的密友兼同事西尔维斯特也是一位数学天才律师，他在 1850 年创造了“矩阵”一词——恰巧，同年，第一条国际电报电缆在多佛和加莱之间铺设。你可以想象到人们兴奋地谈论通过电报将世界统一起来——就像我们最近在互联网的连接能力上看到的创业炒作和真正的兴奋一样。（不幸的是，多佛电缆失败了——但很快其他电缆也失败了。）西尔维斯特和凯莱一样，曾在剑桥学习，但作为犹太人，他没有资格毕业，因为教派规定导致德摩根拒绝在那里任教。因此，西尔维斯特和凯莱以律师的身份相识——但他们都致力于不变理论和他们利用业余时间研究矩阵，最后都成为了数学教授，追寻着自己真正的使命。

Cayley’s close friend and colleague Sylvester, yet another mathematically gifted lawyer, had coined the term “matrix” in 1850—the same year, as it happens, that the first international telegraphic cable was laid, between Dover and Calais. You can imagine the excited talk about uniting the world via telegraphy—the same mix of entrepreneurial hype and genuine excitement we’ve witnessed more recently, over the connective power of the internet. (Unfortunately, the Dover cable failed—but others soon followed.) Sylvester, like Cayley, had studied at Cambridge, but as a Jew he wasn’t entitled to graduate, thanks to the sectarian regulations that had led De Morgan to refuse to teach there. So, Sylvester and Cayley met in their capacity as lawyers—but they both worked on invariant theory and matrices in their spare time, and they both ended up as maths professors, following their true calling.

虽然凯莱被普遍认为是矩阵代数的创始人，但其他数学家很快就开发出了更为复杂的矩阵理论。例如，在美国，父子数学家本杰明和查尔斯·皮尔斯是矩阵和许多其他新后四元数代数的先驱，因为汉密尔顿打开了代数闸门。不过，凯莱寻找矩阵代数规则的动机只是为了更有效地进行计算，这一点很能说明问题：就像在古代一样，许多新的数学被创造出来是为了解决实际问题或促进计算。但是，一旦你可以在一个环境中乘以矩阵，它们就可以像向量一样应用于大量其他实际计算。

While Cayley is generally credited as the founder of matrix algebra, other mathematicians soon developed more sophisticated matrix theory. In America, for example, father and son mathematicians Benjamin and Charles Peirce were pioneers of matrix and many other new, post-quaternion algebras, for Hamilton had opened the algebraic floodgates. Still, it is telling that Cayley had been motivated to find the rules of matrix algebra simply to carry out computations more efficiently: just as in ancient times, a lot of new maths is created to solve practical problems or to facilitate computation. But once you can multiply matrices in one context, they, like vectors, can be applied to an amazing range of other practical computations.

• • •

• • •

说到海量信息，在搜索引擎编程中，可以通过矩阵和向量代数紧凑地处理大量数据。最早的搜索引擎方法之一是使用布尔代数通过布尔的 AND、OR 和 NOT 运算为用户的查询提供精确匹配。但正如康奈尔大学计算机科学教授 Gerry Salton 意识到的那样在 20 世纪 60 年代，矩阵和向量允许部分匹配用户的查询并根据相关性进行排序。（后来，布尔搜索引擎也引入了模糊布尔逻辑来实现这一点，每种方法都有其优点和缺点。）在向量方法中，信息以矩阵的形式存储，行表示关键词，列表示包含这些关键词的不同文档。为了简化问题，我假设数据库只有三个关键词，我将其称为 A、B 和 C。文档 1 可能不包含对关键词 A 和 B 的引用，但包含对 C 的三个引用，因此它将表示为向量$\left(\begin{array}{l}0\\ 0\\ 3\end{array}\right)$，矩阵的第一列。文档 2 可能对 A 有一次引用，对 B 有三次引用，对 C 有两次引用，因此它将表示为$\left(\begin{array}{l}1\\ 3\\ 2\end{array}\right)$等等。现在假设用户想要搜索与 B 相关的信息，那么“查询向量”可以写成$\left(\begin{array}{l}0\\ 1\\ 0\end{array}\right)$. 你用 0 表示它们，而你想要的是 B，用 1 表示。）由于每个文档都用一个向量表示，因此它与查询向量的相关性可以通过标量积（几何定义，如下一个尾注）从其向量相对于查询向量的角度来计算；角度越小，匹配越接近^。20

Speaking of huge amounts of information, in programming search engines, large arrays of data can be handled compactly via matrix and vector algebra. One of the earliest search engine methods used Boolean algebra to give exact matches to a user’s query, via Boole’s AND, OR, and NOT operations. But as Cornell computer science professor Gerry Salton realised in the 1960s, matrices and vectors allow users’ queries to be matched partially and ranked according to relevance. (Fuzzy Boolean logic was later introduced in Boolean search engines to do this, too, and each approach has its advantages and disadvantages.) In the vector approach, information is stored as a matrix, with rows representing, say, key words, and columns representing different documents containing these key words. To simplify matters I’ll assume the database has just three key words, which I’ll call A, B, and C. Document 1 might contain no references to keywords A and B, say, but three references to C, so it would be represented as the vector $\left(\begin{array}{l}0\\ 0\\ 3\end{array}\right)$, the first column of the matrix. Document 2 might have one reference to A, three to B, and two to C, so it would be represented as $\left(\begin{array}{l}1\\ 3\\ 2\end{array}\right)$, and so on. Now suppose the user wants to search for information relating to B, so the “query vector” can be written as $\left(\begin{array}{l}0\\ 1\\ 0\end{array}\right)$. you represent them by 0, and you do want B, denoted by 1.) Since each document is represented by a vector, its relevance to the query vector can be computed, via the scalar product (defined geometrically, as in the next endnote), from its vector’s angle with respect to the query vector; the smaller the angle, the closer the match.²⁰

另外，在谷歌的 PageRank 算法中（该算法由斯坦福大学研究生拉里·佩奇和谢尔盖·布林于 20 世纪 90 年代末开发），网站根据链接到它的其他网站的数量和排名进行分类。表示从网站 A 到其他每个网站的链接比例的向量是矩阵的第一列（我将其称为M），其他网站也是如此。要找到每个网站本身的排名，最好先对所有网站进行相同的排名。然后通过将这个初始排名向量乘以链接偏好矩阵来更新它。然后这个过程不断重复，每次都乘以M，这样到第n步时，初始排名向量已经乘以了M ⁿ。这个过程不断迭代，直到排名向量不再有显著变化。在这个“平衡”点，网站的 PageRank 向量被定义。随着更多网站链接的加入，这个过程不断更新，正如你在搜索某些内容时看到的那样，并注意到哪些网站首先出现。但如果没有矩阵乘法的定义，这一切都不可能实现^。21

Alternatively, in Google’s PageRank algorithm—developed in the late 1990s by Stanford grad students Larry Page and Sergey Brin—a website is classified according to the number and rank of the other websites that link to it. The vector representing the proportionate number of links from website A to each of the other websites is the first column of a matrix (which I’ll call M), and so on for the other websites. To find the rank of each website itself, it makes sense to start off with an equal ranking for all of them. This initial rank vector is then updated by multiplying it by the matrix of linking preferences. The process then keeps repeating, each time multiplying by M, so that by the nth step, the initial rank vector has been multiplied by Mⁿ. The process keeps on iterating until there’s no significant change in the rank vector. At this “equilibrium” point the website’s PageRank vector is defined. With more website links coming on board, the process keeps on updating, as you can see when you search for something and note which websites come up first. But none of this would be possible without a definition of matrix multiplication.²¹

• • •

• • •

在机器学习中，信息也存储在字符串和数组（向量和矩阵，甚至多维数组（实际上是张量的例子））中。在现实世界中，一般只能收集有关某个主题的所有可能数据的样本。数据科学包括从该样本推断以进行预测（比如明年某人心脏病发作的可能性，或在特定时间的可能负荷和能源价格）和分类（比如肿瘤是恶性的还是良性的，电子邮件是有效的还是垃圾邮件，或者最近引起争议的一张脸是否是你的脸）。程序员通过将数据拟合到“模型”中来实现这一点——一种可以理解数据的算法或方程。但有时设计一个适当准确的模型来拟合手头的数据太困难或成本太高，这就是机器学习的作用所在。这个想法是用一种算法对计算机进行编程，通过一个简单的初始模型“训练”计算机进行预测或分类；输入初步“训练”数据、初始模型和错误测试程序。利用向量和矩阵代数，程序员告诉计算机如何估计初始模型的准确性，并不断调整它，直到达到最佳准确性——类似于 PageRank 算法中的矩阵乘法迭代。然后模型就被“训练”了，程序员现在可以使用它根据新数据进行预测。

In machine learning, too, information is stored in strings and arrays— vectors and matrices, and even multidimensional arrays (which are, in fact, examples of tensors). In the real world it is generally only possible to collect a sample of all the possible data about a topic. Data science includes the art of extrapolating from this sample to make predictions—such as the likelihood of a person having a heart attack in the next year, or the likely load and price of energy at a given time—and classifications, such as whether a tumor is malignant or benign, whether an email is valid or spam, or, more recently and controversially, whether a face is yours or not. Programmers do this by fitting the data to a “model”—an algorithm or an equation that makes sense of the data. But sometimes it is too difficult or costly to devise an appropriately accurate model to fit the data at hand, and this is where machine learning comes into it. The idea is to program the computer with an algorithm that “trains” it to make predictions or classifications via a simple initial model; preliminary “training” data are fed in, along with the initial model and an error-testing routine. Making use of vector and matrix algebra, the programmer tells the computer how to estimate the accuracy of the initial model and to keep readjusting it until it reaches an optimum accuracy—analogously to the matrix multiplication iterations in the PageRank algorithm. The model has then been “trained,” and the programmer can now use it to make predictions from new data.

机器学习发展如此迅速，以至于它已经渗透到了日常生活中令人眼花缭乱的各个方面，从聊天机器人和推荐算法到语音和图像识别、欺诈检测、自动驾驶汽车、医疗诊断等等，有时是好的，有时是坏的。（“坏的”不仅包括专制政府或网络犯罪分子对人工智能的滥用，以及对我们生活和工作方式的颠覆性改变，还包括围绕用于训练人工智能的数据的道德问题。这些问题包括未经创建者许可使用数据，以及程序员（主要是白人男性）无意识地将种族、性别和性别偏见编入训练数据、搜索引擎和其他算法中^。22 )

Machine learning has expanded so rapidly that it is now behind a dizzying range of everyday things, from chatbots and recommendation algorithms to speech and image recognition, fraud detection, self-driving cars, medical diagnostics, and much more—for better and sometimes worse. (The “worse” includes not only misuse of AI by repressive governments or cyber criminals, and the disruptive changes to the way we live and work, but also the ethical issues surrounding the data used to train AI. These issues include using data without permission from their creators, and the racial, sexual, and gender biases that programmers—largely white men—have unconsciously programmed into training data, search engines, and other algorithms.²²)

矩阵有无数其他用途——包括密码学，其中矩阵乘法（以及其他代数结构）的非交换性目前正在被探索作为一种额外的安全工具。这项研究的先驱之一是一位 16 岁的爱尔兰女学生莎拉·弗兰纳里。她非常钦佩凯利，以至于她用他的名字命名了她的算法^。23

Matrices are used in countless other ways—including cryptography, where the noncommutativity of matrix multiplication (and of other algebraic structures) is currently being explored as an additional safety tool. One of the pioneers in this research was a sixteen-year-old Irish schoolgirl, Sarah Flannery. She admired Cayley so much that she named her algorithm after him.²³

• • •

• • •

不过，现在，如果你已经学习过线性代数课程，你可能会认为矩阵乘法的一个更简单、更古老、更熟悉的应用是旋转——你是对的！所以，我们开始关注汉密尔顿对旋转的关注。

By now, though, if you’ve taken a linear algebra course, you might be thinking that a simpler, older, and more familiar application of matrix multiplication is rotations—and you are right! So, we are beginning to home in on Hamilton’s preoccupation with rotations.

例如，在机器人技术中，程序员需要知道如何编写指令，以便计算机可以告诉机器人如何从一个地方移动到另一个地方。例如，要抬起和放下机器人手臂，必须绕铰链旋转它（就像肩关节或肘关节一样），如图4.2所示。在计算机图形学中，关键特征的位置向量也可以通过旋转矩阵在空间中移动。

In robotics, for example, programmers need to know how to write instructions so that the computer can tell the robot how to move from one place to another. To raise and lower a robot arm, for instance, it must be rotated about a hinge (just like a shoulder or elbow joint), as in figure 4.2. In computer graphics, too, the position vectors of key features can be moved around in space via rotation matrices.

顺便说一句，如果你用一个没有特征的方形盒子或一个没有标记的球来做实验，你将无法辨别这种非交换性。这是因为在这样的旋转中，球体和立方体会保持它们的形状和方向，因为它们是围绕中心对称的。换句话说，它们在这种旋转序列下是“不变的”。这是凯莱在提出矩阵代数时研究的那种不变量的几何类比。

By the way, if you were to do the experiment with a featureless square box, or a ball with no markings, you wouldn’t be able to discern this noncommutativity. That’s because with rotations like these, spheres and cubes keep their shape and orientation on account of their symmetry about their centres. In other words, they are “invariant” under this sequence of rotations. This is a geometrical analogue of the kind of invariants that Cayley was studying when he came up with matrix algebra.

比凯莱早一个世纪左右，欧拉利用古希腊人开创的球面三角学，对二维和三维旋转数学进行了出色的几何研究。另一方面，矩阵代数则得益于四元数及其非交换乘法——四元数最终使汉密尔顿能够给出空间旋转的第一个纯代数计算。

Almost a century before Cayley, Euler had made a brilliant geometrical study of two- and three-dimensional rotation mathematics, using the spherical trigonometry pioneered by the ancient Greeks. Matrix algebra, on the other hand, is indebted to quaternions and their noncommutative multiplications—and quaternions had enabled Hamilton to give, at last, the very first purely algebraic computations for rotations in space.

首先，为了表示想要旋转的三维矢量，他只取四元数的矢量部分 - 例如，在P = w + ix + jy + kz中设置w = 0 ，这样他就只得到了矢量p = ix + jy + kz。

First up, to represent the three-dimensional vector he wanted to rotate, he took just the vector part of a quaternion—for example, by setting w = 0 in P = w + ix + jy + kz, so that he had just the vector p = ix + jy + kz.

然后，对于进行旋转的代数“机械”，他选择了一个单位四元数U，对所需的旋转轴和角度进行编码。这类似于图 4.3中的单位复数

Then, for the algebraic “machinery” that will do the rotating, he chose a unit quaternion U encoding the required axis and angle of rotation. It’s analogous to the way that in figure 4.3, the unit complex number

b = 余弦 α + i正弦 α

b = cos α + i sin α

是将直线OA绕原点在二维复平面上旋转的“机制”。类似地，要在三维空间中进行旋转，必须将三维向量p乘以四维四元数U——因此，汉密尔顿耐心寻找将i、j和k相乘的正确规则终于有了回报。其余机制都已到位——我已经下一个尾注中描述了这一点——一些巧妙但简单的计算给出了旋转后的向量。尾注显示了当p绕i轴旋转时如何做到这一点，你可以在图 4.4^中看到结果。24

is the “machinery” that rotates the line OA about the origin in the 2-D complex plane. Similarly, to get a rotation in three-dimensional space, the 3-D vector p has to be multiplied by the 4-D quaternion U—so this is where Hamilton’s patient search for the right rules for multiplying his i, j, and k finally paid off. With the rest of the machinery in place—which I’ve described in the next endnote—some nifty but simple calculations give the rotated vector. The endnote shows how to do this when rotating p about the i-axis, and you can see the result in figure 4.4.²⁴

在数学课上，您可能会发现使用矩阵的旋转向量，正如我所提到的，这是一种很好的方法。但事实证明，在更复杂的情况下，四元数比矩阵更有效，因为您需要组合围绕不同轴的几次旋转，以便以特定方式定位物体或通过平滑的旋​​转序列来驱动它 - 例如卫星、飞机、机器人、手机屏幕或计算机动画。四元数比矩阵更紧凑，而且它们更快，因为它们需要的计算更少，并且消耗的计算机功率更少。

In maths classes, you would likely have found a rotated vector using matrices, which is a good way to do it, as I’ve mentioned. But it turns out that quaternions are more efficient than matrices in more complicated situations where you need to combine several rotations around different axes, in order to orient an object in a particular way or to drive it via a smooth sequence of rotations—a satellite, for instance, or an airplane, a robot, your mobile phone screen, or a computer animation. Quaternions are more compact than matrices, and they are faster, because they require fewer calculations and use less computer power.

例如，在对飞机或航天器进行建模、模拟、跟踪或引导时，其方向（或“姿态”）是从三个垂直轴测量的——通常，沿着飞行器的长度、宽度或翼展以及垂直上下方向，原点位于重心。围绕这些轴的旋转分别称为“滚动”、“俯仰”和和“偏航”，它们由操纵杆、油门和方向舵（粗略地说）机械地实现。为了编写模拟、跟踪和电子制导系统程序，绕三个轴的旋转可以表示为三个 3×3 旋转矩阵（图 4.2中给出的示例的 3-D 版本），这些矩阵组合起来即可得出飞行器所需的方向。例如，要调整偏航角和俯仰角，需要将两个相关矩阵相乘 - 如果计算出两个 3×3 矩阵相乘的所有行列步骤，则需要 27 次乘法和 18 次加法。但是，如果这些旋转用四元数而不是矩阵和滚转、俯仰、偏航角来表示，则只有 16 次乘法和 12 次加法。

For example, in modeling, simulating, tracking, or guiding an aircraft or spacecraft, its orientation (or “attitude”) is measured from three perpendicular axes—typically, along the length of the craft, across the width or wingspan, and the vertical up-and-down direction, with the origin at the centre of gravity. Rotations around these axes are called, respectively, “roll,” “pitch,” and “yaw,” and they are implemented mechanically by the yoke, throttle, and rudder (roughly speaking). To program simulations and tracking and electronic guidance systems, rotations about the three axes can be represented as three 3 × 3 rotation matrices—3-D versions of the example given in figure 4.2—and these are combined to give the required orientation of the craft. For instance, to adjust the yaw and pitch angles, you’d multiply the two relevant matrices—and if you count out all the row-by-column steps in multiplying two 3 × 3 matrices, you’ll find there twenty-seven multiplications and eighteen additions. But if these rotations are represented by quaternions instead of matrices and roll, pitch, yaw angles, it turns out that there are only sixteen multiplications and twelve additions.

但这还没有结束。与矩阵不同，四元数没有“万向节锁”问题，该问题曾困扰过阿波罗 11 号登月任务。万向节是一组环，传统上用于在运动过程中保持陀螺仪和其他仪器的稳定；一共有三个环，每个环围绕不同的轴旋转 - 但当其中两个对齐时，只有两个轴可用于定位飞行器。这在数学上反映为，当相关的旋转矩阵相乘时，最终会得到一个具有太多零的方向矩阵，因此它不再包含绕缺失轴旋转的数据。25^因此，计算机会失去对第三维度方向的跟踪。四元数表示不会发生这种情况。

This is not the end of it, though. Unlike matrices, quaternions don’t have the problem of “gimbal lock,” which famously plagued the Apollo 11 mission to the moon. Gimbals are a set of rings that traditionally kept gyroscopes and other instruments steady during motion; there are three rings, each rotating around a different axis—but when two of them align, there are only two axes available for orienting the craft. This is reflected mathematically when the relevant rotation matrices are multiplied and you end up with an orientation matrix with too many zeroes, so that it can no longer include data for rotations about the missing axis.²⁵ So, the computer loses track of the orientation in the third dimension. This doesn’t happen with quaternion representations.

然而，直到最近，四元数才在当今世界占据了一席之地。例如，NASA 于 1981 年首次将其常规用于编程制导、导航和控制系统中的旋转。他们的软件包括矢量、矩阵和四元数代数——因为它们都非常有用，不同的问题适合不同的方法。四元数在航天器定位和引导方面特别有用——例如，它们在模拟登月任务的轨道和跟踪 NASA 的火星探测器方面发挥了重要作用——并且在顺利处理从其收集的原始数据中生成的计算机图像方面也发挥了重要作用。

Yet it is only relatively recently that quaternions have found their place in today’s world. For instance, NASA first used them routinely to program rotations in guidance, navigation, and control systems in 1981. Their software includes vector, matrix, and quaternion algebra—for it is all marvelously useful, and different problems suit different approaches. Quaternions are especially useful both in orienting and guiding a spacecraft—for instance, they played an important role in simulating orbits for missions to the moon, and in tracking NASA’s Mars Exploration Rovers—and in smoothly processing computer images from the raw data it collects.

• • •

• • •

除了这些技术应用之外，四元数旋转还有一些特殊之处——神秘而又令人惊奇。它们具有意想不到的性质，与描述我们世界基本构件（电子、质子和中子）的数学相同。即使你不想阅读图 4.4中旋转矢量计算的尾注，也请注意旋转矢量a表达式中的e ^{i 2θ项。这很令人惊讶，因为它是通过将原始矢量}p乘以单位四元数U = e ^{i θ}得出的。这是四元数旋转代数的一个怪癖，将 θ 变成 2θ。量子力学的一个怪癖是，虽然你可以将普通物质（例如球或行星）旋转 360° 并回到起点，但量子数学表明，电子等粒子需要“旋转” 720° 才能回到其原始状态。

In addition to all these technological applications, there’s something else that makes quaternion rotations special—something mysterious and utterly surprising. They have an unexpected property that they share with the maths describing the very building blocks of our world—electrons, protons, and neutrons. Even if you didn’t feel like reading the endnote on the calculations leading to the rotated vector in figure 4.4, notice in the caption the e^i2θ term in the expression for the rotated vector a. This is surprising, because it was found by multiplying the original vector p by the unit quaternion U = e^iθ. It’s a quirk of quaternion rotation algebra that turns θ into 2θ. And it’s a quirk of quantum mechanics that although you can rotate an ordinary material thing, such as a ball or a planet, through 360° and get it back to its starting point, quantum maths says that a particle such as an electron needs to “spin” through 720° to get back to its original state.

电子并不是像地球或板球那样绕着自身旋转——根据大多数物理学家的说法，最好将电子想象成点，因此它们没有轴可以绕着旋转。但它们具有角动量，就好像它们在旋转一样。这种奇异而出乎意料的现象是通过几年来巧妙地将两个和两个结合起来发现的，正是这种聪明的猜测和耐心的侦查工作让科学如此激动人心。所以，我将花一两页来追踪这一非凡发现的踪迹，因为它还将带我们更深入地了解量子力学和四元数之间迷人的联系。

Not that the electron is actually spinning about itself, like Earth or a cricket ball—according to most physicists, electrons are best imagined as points, so they have no axis to spin around. Yet they have an angular momentum as if they were spinning. This bizarre and unexpected phenomenon was discovered through some ingenious putting of two and two together over several years, the kind of brilliant guesses and patient detective work that makes science so thrilling. So, I’m going to take a page or two to follow the trail of this extraordinary discovery, for it will also take us deeper into the fascinating connection between quantum mechanics and quaternions.

斯特恩的资历包括十年前成为爱因斯坦的第一位博士后学生，但现在他和格拉赫在吸引足够的实验资金方面遇到了麻烦——20 世纪 20 年代初和第一次世界大战之后，德国局势仍然动荡不安。然后他们的设备不断出现故障，直到经过一年的努力，他们才认为自己已经拍摄到了银原子偏转光束的足够好的图像，并将其捕捉到照相屏幕上。但他们并没有得到普通物理学所期望的结果，这表明蒸发原子的角动量的空间方向将是随机分布的，因为它们的方向在加热过程中会随机波动。事实上，他们一开始并没有得到什么结果。但当他们失望地盯着盘子时，两条黑线开始出现，就像某种“隐形墨水”魔法一样。显然，斯特恩抽的廉价雪茄含有大量的硫，当他检查盘子时，他呼出的硫与盘子上看不见的银原子混合，形成了黑色的硫化银！这两条黑线表明银原子角动量的分量只有两个可能的值。换句话说，角动量的空间取向是量子化的。

Stern’s credentials included having been Einstein’s first postdoctoral student a decade earlier, but now he and Gerlach were having trouble attracting sufficient funding for their experiment—in the early 1920s and the wake of World War I, things were still volatile in Germany. Then their equipment kept breaking down, until finally, after a year of struggle, they reckoned they’d taken a sufficiently good image of the deflected beam of silver atoms, which they’d captured on a photographic screen. But they didn’t get the results you’d expect from ordinary physics, which suggested that the spatial orientations of the vaporised atoms’ angular momenta would be randomly distributed, because of random fluctuations of their orientations during the heating process. In fact, they didn’t get much of a result at all, at first. But as they stared at the plate in disappointment, two black lines began to appear, as if by some kind of “invisible ink” magic. Apparently, Stern’s cheap brand of cigars had a high sulfur content, and as he was examining the plate, the sulfur in his breath mixed with the invisible silver atoms on the plate to produce black silver sulfide! These two black lines showed that there were just two possible values for the components of the silver atoms’ angular momenta. In other words, the spatial orientation of angular momentum was quantised.

他们并不完全惊讶于发现这样的结果——一个离散的而非连续的可能值范围——因为尼尔斯·玻尔的早期原子模型是量子化的。然而，随着量子力学的发展，玻尔模型的局限性逐渐显现出来，因此斯特恩和格拉赫到底发现了什么并不清楚。

They weren’t entirely surprised to find something like this—a discrete rather than continuous range of possible values—because Niels Bohr’s early model of the atom was quantised. As quantum mechanics developed, however, limitations in Bohr’s model came to light, so it wasn’t clear just what Stern and Gerlach had found.

随后，两位年轻的荷兰物理学家乔治·乌伦贝克和塞缪尔·古德斯米特提出，斯特恩和格拉赫测量的可能不是轨道角动量，而是自旋角动量——就好像产生磁场的电子在围绕自身旋转一样。他们通过跟踪氢原子光谱中的一些异常现象得出了这个想法，氢原子每个原子只有一个电子。26^乌伦贝克和古德斯米特利用磁场操纵电子的能级，发现如果电子的角动量仅取决于两个可能的独立值，那么得到的谱线将符合这一自旋假设——这正是斯特恩和格拉赫发现的^。27

Then two young Dutch physicists, George Uhlenbeck and Samuel Goudsmit, suggested, in effect, that perhaps what Stern and Gerlach had measured were not orbital angular momenta but spin angular momenta—as if the electrons generating the magnetic fields were spinning around themselves. They’d come up with this idea by following up some anomalies in the atomic spectrum of hydrogen, whose atoms each have just one electron.²⁶ Using magnetic fields to manipulate the energy levels of the electrons, Uhlenbeck and Goudsmit figured out that the resulting spectral lines would fit this spin hypothesis if the electron’s angular momentum depended on just two possible independent values—which is just what Stern and Gerlach had found.²⁷

当乌伦贝克与荷兰传奇物理学家亨德里克·洛伦兹（Hendrik Lorentz）谈论这个奇怪的想法时（洛伦兹将在后面的相对论中提到），洛伦兹证实了他的担忧，他说，如果产生这种动量的电子真的在旋转，那么它的速度一定高得不可思议。所以，电子并不是像板球一样旋转——它只是表现得像在旋转一样。这看起来太奇怪了，乌伦贝克绝望地拜访了他和古德斯米特的导师保罗·埃伦费斯特（Paul Ehrenfest），因为他们已经给他寄了一篇关于他们假设的论文。但埃伦费斯特很快就把它寄出去发表，告诉惊慌失措的乌伦贝克，“好吧，这是一个好主意，尽管它可能是错的。但你没有名气，所以你没什么可失去的。”这对年轻的乌伦贝克来说并没有多大安慰^。28

When Uhlenbeck talked about this strange idea with the legendary Dutch physicist Hendrik Lorentz—who will come into this story later in connection with relativity—Lorentz confirmed his fears, saying that if the electron producing this momentum really were spinning, it would have to be doing so at an impossibly high speed. So, the electron isn’t spinning like a cricket ball—it just acts as if it is. Which seemed so bizarre that Uhlenbeck made a desperate visit to his and Goudsmit’s mentor, Paul Ehrenfest, for they’d already sent him a paper about their hypothesis. But Ehrenfest had promptly sent it off for publication, telling the panic-stricken Uhlenbeck, “Well, it’s a nice idea, though it may be wrong. But you don’t have a reputation, so you have nothing to lose.” Which wasn’t much comfort for young Uhlenbeck.²⁸

当古怪的英国物理学家保罗·狄拉克开始发展电子行为的相对论量子力学描述时，一切都变得清晰起来。他发现，要使电子的波动方程具有相对论性，他必须加入一些量，这些量刚好给出乌伦贝克和古德斯米特发现的自旋动量值。德国物理学家沃尔夫冈·泡利已经开始研究自旋的非相对论数学，并引入了三个现在称为泡利矩阵的量，狄拉克在他的公式中采用了这些量。正如泡利明确指出的那样，这三个矩阵——有助于描述围绕三个空间轴的自旋角动量分量——以完全相同的方式相互关联作为汉密尔顿的单位四元数 i、j、k：将其中任意两个相乘，即可得到第三个的正负值——这与汉密尔顿八十五年前在布鲁姆桥上雕刻的规则相同！²⁹

It all came together when the eccentric English physicist Paul Dirac began developing a relativistic quantum mechanical description of electron behavior. He found that to make an electron’s wave equation relativistic, he had to include quantities that gave just the values of spin momentum that Uhlenbeck and Goudsmit had found. The German physicist Wolfgang Pauli had already made a start on the nonrelativistic mathematics of spin and had introduced three quantities now known as Pauli matrices, which Dirac adapted in his formulation. As Pauli spelled out explicitly, these three matrices—which help describe spin angular momentum components about the three spatial axes—relate to each other in exactly the same way as Hamilton’s unit quaternions i, j, k: multiply any two of them and you get plus or minus the third—the same rules Hamilton had carved on Broome Bridge eighty-five years earlier!²⁹

此外，数学表明，如果你想旋转量子粒子的自旋轴，旋转方程类似于四元数旋转的方程。这就是为什么电子——以及物质的其他构成要素——具有这种奇怪的特性，你需要将它们旋转两次才能回到起点^。30

What’s more, the maths says that if you want to rotate the spin axis of a quantum particle, the rotation equations are analogous to those for quaternion rotations. That’s why electrons—and the other building blocks of matter—have that strange property where you need to rotate them twice to get back to where you started.³⁰

不过，正如我所强调的那样，亚原子粒子实际上并不像板球那样旋转；相反，它们的自旋与磁场对齐，因此如果你能慢慢旋转磁场，你就会旋转自旋轴（如果不是粒子本身的话）。这正是实验物理学家在这一惊人数学结果出现半个世纪后成功做到的。1975 年，两位澳大利亚物理学家托尼·克莱因和杰夫·奥帕特进行了一项实验，表明这种奇怪的旋转行为是物理现象——而不仅仅是数学现象。他们的实验与托马斯·杨著名的双缝实验有相似之处，后者表明两束光之间的干涉图案是波的干涉图案，而不是粒子的干涉图案。克莱因和奥帕特将一束中子（以物质波的形式传播）衍射成两部分；然而，在这种情况下，一部分与旋转自旋轴的外部磁场相互作用，另一部分保持不变。这两束光产生的干涉图样表明，自旋必须旋转两个周期才能回到原来的状态。这正是数学所预测的！³¹

As I’ve emphasised, though, subatomic particles don’t really rotate like cricket balls; rather their spin aligns with a magnetic field, so if you could slowly rotate the field, you’d be rotating the spin axis (if not the particle itself ). This is just what experimental physicists succeeded in doing, half a century after this astounding mathematical result. In 1975, two Australian physicists, Tony Klein and Geoff Opat, carried out an experiment that showed that this strange rotational behavior was physical—it wasn’t just a mathematical artifact. Their experiment has some kinship with Thomas Young’s famous two-slit experiment, which showed that the interference pattern between two beams of light was that of waves, not particles. Klein and Opat diffracted a beam of neutrons (traveling as a matter wave) into two parts; in this case, however, one part interacted with an external magnetic field that rotated the spin axis, and the other remained unchanged. The interference pattern made by these two beams showed that the spin had to be rotated through two cycles to get back to its original state. Which is just what the maths predicted!³¹

这是一个令人震惊的结果，但在克莱因和奥帕特将他们的研究成果发表之前，另外两个团队也证实了这个奇怪的数学预测。32^自旋，无论它是什么，绝对是真实存在的，它的行为遵循汉密尔顿在最终设法在三维空间中旋转矢量时发现的相同旋转数学怪癖。

It was a stunning result, but before Klein and Opat could get their work into print, two other teams also confirmed this strange mathematical prediction.³² Spin, whatever it was, was definitely real, and it behaved according to the same quirk of rotational maths that Hamilton found when he finally managed to rotate vectors in three-dimensional space.

如今，“旋转”已成为日常生活中不可或缺的一部分，例如用于医学诊断的磁共振成像 (MRI)。MRI 可拍摄患者内脏器官的 3D 图像，因此无需进行侵入性诊断手术或放射治疗，它通过使用磁场来调整氢原子的自旋，由于我们体内水分和脂肪含量高，氢原子无处不在。然后，这些自旋通过无线电波发生偏转，不同的无线电脉冲序列突显身体的不同部位；当无线电源关闭时，图像就会被捕获，自旋会恢复到平衡状态，并在这一过程中释放出电磁能。

Today “spin” is a vital part of everyday life—in the magnetic resonance imaging (MRI) used for medical diagnoses, for example. MRI takes 3-D images of a patient’s internal organs—so there’s no need for invasive diagnostic surgery or radiation—and it does this by using magnetic fields to align the spins of hydrogen atoms, which are ubiquitous given the amount of water and fat in our bodies. The spins are then deflected via radio waves, different sequences of radio pulses highlighting different parts of the body; images are captured when the radio source is turned off and the spins return to their equilibrium state, giving off electromagnetic energy in the process.

• • •

• • •

如果汉密尔顿知道四元数与电子自旋这一基本特性有关，他一定会欣喜若狂。如果他能看到四元数为地球带来的宇宙的壮丽景象，他那充满诗意的灵魂就会为这一切的奇妙、宏伟和神秘而激动不已——他曾在自己最喜欢的一首诗《月全食颂》中试图捕捉这种神秘。这首诗写于他还不到 18 岁，一直保存到去世。在诗中，他向月亮女神询问月食期间“深红色”笼罩在她“美丽的额头”上的“不祥的阴暗”，问这是由来自另一个世界的云朵还是巫师的魔法造成的？或者，也许是我们地球的“阴影锥”使月亮在“我们好奇的眼睛”眼中变得昏暗？这种诗意的想象与科学的结合是汉密尔顿的典型风格。他很想知道，他的四元数（以及现在被称为哈密顿动力学，量子力学就是以此为基础建立的）是否有助于揭示宇宙中一些最神奇的秘密^。33

Hamilton would be beside himself with joy if he knew that quaternions are related to something as fundamental as this strange electron spin property. And if he could see the glorious pictures of our universe that quaternions have helped bring to Earth, his poetic soul would thrill to the wonder and grandeur and mystery of it all—the kind of mystery he’d tried to capture in one of his favourite poems, Ode to the Moon under Total Eclipse. He’d written it when he was not quite eighteen, and he kept it till the day he died. In the poem, he questions the moon goddess on the “portentous gloom” as “crimson red” spreads over her “lovely brow” during the eclipse, asking if it be caused by some passing cloud from another world, or a wizard’s magic spell? Or, perhaps, is it our Earth’s “shadowy cone” that dims the moon to “our wondering eye?” This mix of poetic imagination and science is typical of Hamilton. He would have loved to know that his quaternions (and also what is now known as Hamiltonian dynamics, on which quantum mechanics was modeled) have helped bring to light some of the universe’s most magical secrets.³³

然而，当他第一次发现四元数时，汉密尔顿只瞥见了它们更广泛的可能性。然而，从一开始，他就深信四元数在物理应用中起着至关重要的作用——尽管在他的一生中，四元数从未完全实现他的期望，从未像今天与 NASA 和量子力学的联系那样引人注目，但他仍然拥有自己的荣耀。例如，1848 年，他因发现四元数而获得了爱尔兰皇家学院和爱丁堡皇家学会的奖章。

When he first discovered quaternions, though, Hamilton only glimpsed their wider possibilities. Yet right from the beginning he felt in his bones that quaternions had a vital role to play in physical applications—and if in his lifetime they never quite lived up to his hopes, never achieving the glamorous profile that they have today by association with NASA and quantum mechanics, still he had his share of glory. In 1848, for example, he was awarded medals for his discovery of quaternions, from both the Royal Irish Academy and the Royal Society of Edinburgh.

他长期致力于理解复数，将其扩展到三维，并发明了四维四元数，这让我们想起了即使是最简单的事情，我们今天也认为理所当然，但理解这些事情需要一种深刻的思考。德摩根和皮科克在符号代数方面的工作也是如此，他们梳理出了人们长期以来一直在使用的代数定律，却从未意识到它们可能适用于数字以外的事物。今天，主要是数学哲学家讨论数字与符号、语言和意义之间的关系。教育政策制定者和资助机构主要关注数学如何发挥作用。因此，值得再次强调的是，汉密尔顿不知道他的四元数在驾驶机器人和航天器等高科技应用中会有多么有用。他也没有预见到向量将在物理、工程和 IT 中发挥多么普遍的作用，因为现在它们几乎出现在你需要指定物体空间位置或存储和处理数据的任何地方。这表明，尽管现代数学源于古代试图解决美索不达米亚人面临的实际问题，尽管解决实际问题的需要仍然是新数学见解的主要驱动力，但有时，为数学而数学却能解决未来的技术和物理问题。

His long struggle to make sense of complex numbers, to expand them to three dimensions and to invent his 4-D quaternions, reminds us of the kind of deep thinking needed to make sense of even the simplest things we take for granted today. The same goes for De Morgan and Peacock’s work on symbolic algebra, teasing out the laws of algebra that people had long been using without ever realising they might apply to things other than just numbers. Today, it is mainly philosophers of mathematics who discuss such things as the relationship between numbers and symbolism, language, and meaning. Educational policymakers and funding bodies are focussed mainly on how mathematics can be useful. So it is worth saying again that Hamilton had no idea just how useful his quaternions would prove in such high-tech applications as driving robots and spacecraft. Nor did he foresee what a ubiquitous role vectors would play in physics, engineering, and IT, for now they crop up just about everywhere you need to specify the spatial positions of objects or to store and handle data. Which goes to show that although modern maths has its ancient roots in attempts to solve the kinds of practical problems the Mesopotamians were faced with—and although the need to solve practical problems is still a key driver of new mathematical insights—sometimes it is maths for maths’s sake that leads to solutions of future technological and physical problems.

事实上，在汉密尔顿告诉他的朋友约翰·格雷夫斯他的四元数的几个月后，格雷夫斯就发明了“八元数”——不是四维的，而是由四元数对组成的八维复数。它们只是一种好奇，证明代数世界除了数字和四元数之外还有更多的东西——然而今天人们正在研究它们与粒子物理标准模型的可能联系。这是当前的模型，其中物理现实的构成要素被分为具有相似属性的类别。例如，物质由不同种类的夸克和轻子组成，后者是带电的粒子，比如电子。还有力的载体或玻色子，比如携带电磁力的光子，以及携带质量的希格斯玻色子。

In fact, a couple of months after Hamilton told his friend John Graves about his quaternions, Graves invented “octonions”—not 4-D but 8-D complex numbers made from pairs of quaternions. They were just a curiosity, proof that there was even more to the algebraic world than numbers and quaternions—yet today they are being investigated for possible connections with the standard model of particle physics. This is the current model in which the building blocks of physical reality are classified in categories that share similar properties. For instance, matter is made of different kinds of quarks and leptons, the latter being particles with charge, such as electrons. Then there are the force-carriers or bosons, such as the photons that carry electromagnetic force, plus the Higgs boson, which carries mass.

目前还不清楚八元数是否会在这种背景下发挥作用——但正如汉密尔顿非交换乘法的应用一样，一旦一个想法被提出，你就永远不会不知道它会激发什么。34^乔治·布尔是另一个例子；他的代数逻辑系统是从皮科克和德·摩根奠定的符号代数基础以及汉密尔顿的四元数发展而来的。布尔在一场倾盆大雨中感冒后突然去世，当时他还太年轻，但他并不知道，他用 0 和 1 来表达真假陈述的系统最终会驱动我们的计算机和其他电子设备和小工具。

The jury is still out on whether or not octonions will prove useful in this context—but as with the applications that flowed from Hamilton’s noncommutative multiplication, once an idea is out there, you never know what it may inspire.³⁴ George Boole is another example; his system of algebraic logic evolved from the foundations of symbolic algebra laid by Peacock and De Morgan, and by Hamilton’s quaternions. Little did Boole know, when he died suddenly and far too young after catching cold in a downpour of rain, that his system of expressing true and false statements in terms of 0s and 1s would eventually drive our computers and other electronic devices and gadgets.

然而，早在 19 世纪 40 年代，矢量分析的发展还有很长的路要走，它不仅拓展了我们的技术，还拓展​​了我们对数学和宇宙的理解。在接下来的几章中，我们将遇到新一代数学家和数学物理学家，他们开发汉密尔顿矢量和四元数的工作将带我们进入更加奇妙的现实领域。

Back in the 1840s, though, there was still a way to go in developing the vector analysis that has broadened not just our technology but also our understanding of maths, and of our universe. Over the next few chapters, we’ll meet new generations of mathematicians and mathematical physicists, whose work developing Hamilton’s vectors and quaternions will take us into yet more wondrous realms of reality.

不过，首先，我们会看到，新想法总是需要时间来证明其价值。我们还会遇到一位非凡的局外人，他发明了自己的强大矢量系统。他的动机与汉密尔顿不同——他对旋转和复数不感兴趣——但最终，我们会看到，这两种方法都有助于数学家开发当今如此重要的复杂矢量和张量分析。

First, though, we’ll see that it always takes time for new ideas to prove their worth. And we’ll meet an extraordinary outsider, who invented his own powerful vectorial system. His motivation was different from Hamilton’s—he wasn’t interested in rotations and complex numbers—but ultimately, we’ll see that both approaches would help mathematicians to develop the sophisticated vector and tensor analyses that are so important today.

如今我们知道汉密尔顿的四元数是多么有用，但在当时，它们面临着一场激烈的争论。1847 年，汉密尔顿向他的朋友罗伯特·格雷夫斯幽默地讲述了他在英国科学促进会会议上发表关于四元数的论文后讨论的情况。顺便说一句，正是在 1833 年的一次英国协会会议上，“科学家”一词首次被引入，目的是将科学和数学研究人员置于与艺术家相同的专业地位。这个想法是威廉·惠威尔提出的，他还在 1840 年创造了“物理学家”这个名字。无论如何，1847 年，汉密尔顿告诉格雷夫斯，虽然皮科克博士认为他已经“充分阐述”了他的四元数系统，但另一位听众却怀疑“使用我的新微积分时可能会犯错误。作为回应，我否认有权对犯错的能力设置任何限制。” ¹

These days we know how useful Hamilton’s quaternions are, but at the time they faced quite a battle for acceptance. In 1847, Hamilton gave his friend Robert Graves a humorous account of the discussion that followed his presentation of a paper on the subject, at a meeting of the British Association for the Advancement of Science. Incidentally, it was at a British Association meeting in 1833 that the term “scientist” was first introduced, in order to put scientific and mathematical researchers on the same professional footing as artists. The idea was William Whewell’s, and he also coined the name “physicist” in 1840. Anyway, in 1847 Hamilton told Graves that while Dr. Peacock thought he’d given “a capital exposition” of his quaternion system, another member of the audience wondered about “the possibility of making mistakes in the use of my new calculus. In reply to which I disclaimed the power of setting any limit to the faculty of making blunders.”¹

汉密尔顿继续说，这种“温和的小冲突”还有几次，然后博学家约翰·赫歇尔爵士给四元数带来了狂热的认可。这对皇家天文学家乔治·艾里来说太过分了。艾里曾称汉密尔顿对圆锥折射的数学预测（我们在第 2 章中看到）“可能是有史以来最了不起的预测”——但他对四元数代数的看法要悲观得多。汉密尔顿告诉格雷夫斯，赫歇尔发言后，艾里站起来“讲述他自己对 [四元数] 的了解，他承认自己根本不了解四元数；但 [他] 让我们明白，他不知道的东西不值得知道。”换句话说，汉密尔顿讽刺地补充道，艾里似乎认为任何在他看来晦涩难懂或自相矛盾的东西一定是错误的^。2

There were a few more “gentle skirmishes” of this sort, continued Hamilton, and then the polymath Sir John Herschel gave quaternions a rapturous endorsement. Which was too much for the Astronomer Royal, George Airy. Airy had called Hamilton’s mathematical prediction of conical refraction (which we saw in chap. 2) “perhaps the most remarkable prediction that has ever been made”—but he took a much dimmer view of quaternion algebra. Hamilton told Graves that after Herschel spoke, Airy rose to his feet “to speak of his own acquaintance with [quaternions], which he avowed to be none at all; but [he] gave us to understand that what he did not know could not be worth knowing.” In other words, Hamilton added wryly, Airy seemed to assume that anything that to him seemed obscure or paradoxical must be erroneous.²

如果像汉密尔顿这样地位的科学家都难以让自己的系统被接受，想象一下，对于一个生活在偏远乡村小镇的自学成才的孤独者来说，这将会有多困难。空气中一定有某种矢量，因为这样的人确实存在：德国教师赫尔曼·格拉斯曼。在科学界经常发生的独立平行发现中，格拉斯曼发现了一个与汉密尔顿的系统类似的系统，这是其中的一个非凡例子。其他几个人——尤其是因“莫比乌斯带”而出名的德国人奥古斯特·莫比乌斯和意大利人朱斯托·贝拉维蒂斯——也一直在向这个想法靠拢。但最终成功的是汉密尔顿和格拉斯曼——两人都不知道对方是谁。

If someone of Hamilton’s scientific standing was having trouble getting his system accepted, imagine how much harder it would be for a self-taught loner living in a remote country town. There must have been something vectorial in the air, for such a person actually existed: the German schoolteacher Hermann Grassmann. In one of those remarkable instances of independent parallel discovery that happen surprisingly often in science, Grassmann discovered a system analogous to Hamilton’s at the very same time. Several others, too—notably the German August Möbius, of “Möbius strip” fame, and the Italian Giusto Bellavitis—had also been inching toward the idea. But it was Hamilton and Grassmann who ultimately succeeded—each unbeknownst to the other.

格拉斯曼比汉密尔顿小四岁，他的父亲是波美拉尼亚神学家尤斯图斯·格拉斯曼，他在斯德丁（现称什切钦，因为该地区位于今天的波兰西北部）的当地高中教授数学和科学。赫尔曼和他的十一个兄弟姐妹都出生在斯德丁，但在学校里，他没有表现出任何数学天赋，不像神童汉密尔顿——尤斯图斯认为他的儿子可能会找到一份园丁或工匠的工作。尽管如此，赫尔曼还是设法考上了柏林大学，在那里学习了语言学和神学——不管他父亲怎么想，他对成为一名园丁并不感兴趣。事实上，他的目标是成为一名像尤斯图斯一样的教师，所以当他从大学回家后，他开始自学数学和科学。他的父亲编写了一些数学教科书，显然这些教科书效果很好，因为赫尔曼很快就通过了国家教师考试^。3

Four years younger than Hamilton, Grassmann was the son of a Pomeranian theologian, Justus Grassmann, who taught maths and science at the local high school in Stettin—now spelled Szcezcin, because the area is in today’s northwest Poland. Hermann and his eleven siblings were born in Stettin, but at school he’d shown no mathematical talent at all, unlike the prodigy Hamilton—Justus thought his son might find a job as a gardener or craftsman. Still, Hermann managed to get into Berlin University, where he studied philology and theology—he wasn’t interested in becoming a gardener, no matter what his father thought. In fact, he’d set his eyes on becoming a teacher like Justus, so when he returned home from university, he began to study mathematics and science on his own. His father had written some maths textbooks, and they evidently worked well, for soon Hermann passed the state teaching exams.³

在柏林教了一年书之后，他便在家乡教授高中数学、科学、语言和神学，并度过了余生。与同样自学数学，后来当了多年教师的乔治·布尔不同，赫尔曼·格拉斯曼从未获得同行的认可，因此无法获得大学职位。这当然不是因为缺乏才华：格拉斯曼在学校可能没有表现出多少天赋，但他是一位才华横溢的独创数学家。这也不是因为他缺乏努力——他非常有野心，并付出了非凡的努力来争取职业晋升。

After a year teaching in Berlin, he taught high school maths, science, languages, and theology in his hometown for the rest of his life. Unlike George Boole, who also taught himself mathematics and then spent years as a schoolteacher, Hermann Grassmann never gained the recognition from his peers that might have led to a university position. It certainly wasn’t for lack of merit: Grassmann may not have shown much talent at school, but he was a brilliantly original mathematician. And it wasn’t for want of trying, either—he was quite ambitious, and put in extraordinary efforts to try to gain professional promotions.

他开始通过深入研究潮汐来证明自己，并在此过程中意识到，就像牛顿一样，空间中的物理现象是矢量的（他们俩都没有使用这个术语）。但是，虽然牛顿只使用了平行四边形规则以及力和速度既有大小又有方向的思想，但格拉斯曼却像汉密尔顿一样着迷于发展矢量代数的底层数学的挑战。当他将这种代数应用于潮汐理论时，他发现自己“对这种方法得出的计算结果的简单性感到震惊” ^。4

He began trying to prove himself with an intensive study of the tides, and in the process he realised, like Newton, that physical phenomena acting in space are vectorial (not that either of them used that term). But while Newton used just the parallelogram rule and the idea that forces and velocities have both magnitude and direction, Grassmann became as entranced as Hamilton by the challenge of developing the underlying mathematics of vector algebra. And when he applied this algebra to the theory of tides, he found himself “astounded by the simplicity of the calculations resulting from this method.”⁴

1840 年，格拉斯曼准备将自己长达 200 页的潮汐理论论文提交给决定教师晋升的柏林考试委员会。不幸的是，但也许并不奇怪，考官未能认识到论文革命性的数学基础。格拉斯曼一定有非凡的自信，独自工作，无人认可他的成就。但他坚信新方法能够简化计算，因此，尽管关于潮汐的论文被拒，以及因此无法晋升，他似乎并没有气馁，在 1844 年，他发表了一篇关于新方法的巨作。他把它命名为Die Lineale Ausdehnungslehre，即“线性扩展理论”。

By 1840, he was ready to submit his 200-page dissertation on tidal theory to the Berlin examination committee that decided on teacher promotions. Unfortunately, but perhaps not surprisingly, the examiner failed to appreciate its revolutionary mathematical basis. Grassmann must have had extraordinary self-belief, working away in isolation, with no one to recognise his achievements. But he passionately believed in the power of his new method to simplify computations, and so, seemingly undaunted by the rejection of his paper on the tides—and by the consequent lack of promotion up the teaching ladder—in 1844 he published a monumental treatise on his new approach. He called it Die Lineale Ausdehnungslehre, or “The Linear Theory of Extensions.”

这是一种冗长的说法，表明他的几何“线”的乘法不交换——这表明皮科克和德·摩根在识别和命名算术基本定律方面的工作在概念和语言上具有重要意义。事实上，格拉斯曼关于将 + 改为 − 的说法，是一种表明向量积是反交换的。这正是叉积的右手定则所显示的（正如我们在图 4.1中看到的）：如果你将手指从p弯曲到q，拇指指向上方，那么如果你将它们从另一个方向弯曲到p，你的拇指就会指向下方——这意味着p × q = − q × p。

This was a long-winded way of saying that multiplication of his geometric “lines” was not commutative—which goes to show the conceptual and linguistic importance of Peacock and De Morgan’s work on identifying and naming the fundamental laws of arithmetic. In fact, Grassmann’s talk of changing + to − was a way of saying that vector products are anticommutative. This is just what the right-hand rule for cross products shows (as we saw in fig. 4.1): if you curl your fingers from p to q and your thumb points upward, then if you curl them the other way, from q to p, your thumb will point downward—which means that p × q = −q × p.

我之前提到过，年轻的汉密尔顿研究过拉普拉斯的《天体力学》，发现平行四边形规则的逻辑存在错误，而他之所以发现四元数，是因为他对朋友德摩根的符号代数工作和沃伦的复数几何论文很感兴趣。格拉斯曼也知道平行四边形规则，但除此之外，他通过一条非常不同的途径得出了他的“扩展”理论。

I mentioned earlier that the young Hamilton had studied Laplace’s Mécanique Céleste and found an error in the logic of the parallelogram rule, and that he had discovered quaternions because he’d been intrigued by his friend De Morgan’s work on symbolic algebra and by Warren’s paper on the geometry of complex numbers. Grassmann also knew about the parallelogram rule, but otherwise he came to his theory of “extensions” by a very different route.

他从意大利裔法国数学家约瑟夫-路易·拉格朗日的一本教科书中了解到代数和微积分在物理学中的力量。这本书名为《分析力学》，它曾被拉普拉斯的《天体力学》问世于半个世纪前——1788 年，即法国大革命前夕；而拉普拉斯的《天体力学》问世则晚了十年。拉格朗日和拉普拉斯对政治并不特别感兴趣，在革命年代里设法保持中立。1790 年，法国科学院成立了度量衡委员会——拉普拉斯和拉格朗日都是委员会成员——但当 1793 年恐怖统治开始时，狂热的新政府关闭了科学院。他们还算明智，让度量衡委员会继续运作——除了一些他们认为可疑的成员，包括外国人^6——正是由于这个委员会，公制才得以创立。

He learned about the power of algebra and calculus in physics from a textbook by the Italian-French mathematician Joseph-Louis Lagrange. It was called Mécanique Analytique (Analytical Mechanics), and it had been published half a century earlier—in 1788, the eve of the French Revolution; Laplace’s Celestial Mechanics appeared a decade later. Lagrange and Laplace were not particularly interested in politics and managed to steer a neutral course through the Revolutionary years. In 1790 the French Academy of Science created a Committee on Weights and Measures—both Laplace and Lagrange were members—but when the Reign of Terror began in 1793, the fanatical new government closed the Academy. They did have the sense to keep the weights and measures committee going—minus a few members they deemed suspect, including foreigners⁶—and it is thanks to this committee that the metric system was created.

至于《分析力学》，拉普拉斯更新了牛顿关于行星运动的著作——格拉斯曼以拉普拉斯为基础提出了潮汐理论——而拉格朗日则扩展了牛顿的一般运动概念。我提到约翰·伯努利、埃米莉·杜·夏特莱，尤其是莱昂哈德·欧拉已经开始将牛顿的著作翻译成莱布尼茨微积分的语言，但拉格朗日扩展了微分方程的理论，使它们最终成为力学的语言——即我们今天习以为常的运动和力的语言。拉格朗日是父母十一个孩子中唯一一个活过童年的人，他确实让他们感到骄傲^。7

As for Mécanique Analytique, while Laplace had updated Newton’s work on planetary motion—Grassmann built on Laplace for his theory of tides—Lagrange extended Newton’s concept of motion in general. I mentioned that Johann Bernoulli, Émilie du Châtelet, and especially Leonhard Euler had begun translating Newton’s work into the language of Leibnizian calculus, but Lagrange extended the theory of differential equations so that they finally became the language of mechanics—the language of motion and forces that we take for granted today. Lagrange was the only one of his parents’ eleven children to survive beyond childhood, and he certainly did them proud.⁷

格拉斯曼在《解析学》的前言中承认自己受益于《分析力学》 ——但他指出，使用自己的新系统，“计算结果往往比拉格朗日的工作短 10 倍以上。”他指的是，如果使用四元数进行复合旋转，表达式的简洁性也类似。然而，格拉斯曼的向量之旅并非始于拉格朗日，而是始于他父亲的教科书——拥有这样的父亲是他长期努力争取数学成功的少数幸运之一。

Grassmann acknowledged his debt to Mécanique Analytique in the foreword of Ausdehnungslehre—but he noted that in using his own new system, “the calculations often came out more than ten times shorter than in Lagrange’s work.” What he meant was a similar economy of expression as you get if you use quaternions for compound rotations. Grassmann’s journey to vectors had begun not with Lagrange, however, but with his father’s textbooks—and having such a father was one of the few pieces of luck he had in his long struggle for mathematical success.

正如他在《几何学》的前言中指出的那样，他的父亲将矩形定义为其长度和宽度的“几何乘积”。这本身就很新颖——你可以将两个几何量相乘来创建一个新的几何对象——在这种情况下，“乘以”两条线得到一个矩形。相比之下，这种传统的乘积上下文将表示长度和宽度的数字相乘得到另一个数字，古人很久以前就将其定义为矩形的面积。但赫尔曼将他父亲的想法带入了真正的矢量领域，他指出，如果你把边“不仅仅是长度，而是有向量”的话，你可以对所有平行四边形做同样的事情。当然，“有向量”是汉密尔顿“矢量”的另一个名字。

As he noted in the foreword to Ausdehnungslehre, his father had defined a rectangle as the “geometrical product” of its length and width. This was novel in itself—the idea that you could multiply two geometric quantities in themselves to create a new geometric object—in this case, “multiplying” two lines to get a rectangle. By contrast, the traditional product in this context multiplies the numbers representing the length and width to get another number, which the ancients had long ago defined as the area of the rectangle. But Hermann took his father’s idea into true vectorial territory by noting that you could do the same for all parallelograms, if you viewed the sides “not merely as lengths, but rather as directed magnitudes.” And, of course, “directed magnitude” is another name for Hamilton’s “vector.”

您可能还记得，向量积a × b的模的几何定义是 | a || b |sin θ，其中 | a | 表示向量a的模，| b |也是同样如此。这就是为什么还有另一个关于叉积的笑话（来自电视情景喜剧《班长》）：“将大象和葡萄放在一起会得到什么？”答案是：“大象 葡萄 sine-theta！”更严肃地说，对于矩形，θ = 90°，因此 sin θ = 1，这意味着您又回到了传统规则，即只需将边的模相乘即可找到面积。换句话说，sin θ 的作用对于矩形是隐藏的，但对于其他平行四边形则不是。

You might remember from school that the geometric definition of the magnitude of the vector product a × b is |a||b|sin θ, where |a| means the magnitude of the vector a, and similarly for |b|. That’s why there’s yet another cross-product joke (from the television sitcom Head of the Class): “What do you get when you cross an elephant and a grape?” Answer: “Elephant grape sine-theta!” More seriously, in the case of a rectangle, θ = 90°, and so sin θ = 1, which means you’re back to the traditional rule where you find the area simply by multiplying the magnitudes of the sides. In other words, the role of sin θ is hidden for rectangles, but not for other parallelograms.

格拉斯曼将这种几何乘法称为“外积”。在三维空间中，它的解释与汉密尔顿（和现代）的矢量积略有不同，如您在尾注中看到的那样，但在计算上是相同的。8^格拉斯曼还定义了我们（遵循汉密尔顿）所说的两个矢量的“标量积”，但他称之为“内积”。他使用您可能已经学过的标量积的相同几何定义来定义他的内积，a ∙ b = | a || b |cos θ——尽管他当然没有这种现代矢量符号。

Grassmann called this kind of geometrical multiplication an “outer product.” In three dimensions, it differs slightly in its interpretation from Hamilton’s (and the modern) vector product, as you can see in the endnote, but computationally it is the same.⁸ Grassmann also defined what we, following Hamilton, call the “scalar product” of two vectors, but which he called an “inner product.” He defined his inner product using just the same geometrical definition you may have learned for the scalar product, a ∙ b = |a||b|cos θ—although he didn’t have this modern vectorial notation, of course.

虽然我们今天在向量分析中使用汉密尔顿的术语“向量”和“标量”，但我们稍后会看到，格拉斯曼的“内”和“外”积最终也会找到现代的归宿，分别作为n维向量和张量的乘积的名称。汉密尔顿曾告诉德摩根_，他认为没有理由找不到他称之为“多元组”（与“三元组”类似）的代数，形式为 ( _a1 、 a2 、 …、an ₎，但他止步于四元数，其中n = 4（或当重点放在向量部分时为 3）。格拉斯曼的方法也侧重于三维，但他以一种抽象的方式呈现了它，这种方式很容易适应任意维数——事实上，他应用它开创了n维线性代数的新理论。

While we use Hamilton’s terms “vector” and “scalar” in vector analysis today, we’ll see later that Grassmann’s “inner” and “outer” products would also eventually find a modern home, as the names for products of n-dimensional vectors and tensors, respectively. Hamilton had told De Morgan he saw no reason you couldn’t find an algebra of what he called “polyplets” (by analogy with “triplets”) of the form (a₁, a₂, … , a_n), but he had stopped with quaternions, where n = 4 (or 3 when the focus is on the vector part). Grassmann’s approach also focussed on three dimensions, but he presented it in an abstract way that was readily adaptable to any number of dimensions—in fact, he applied it to pioneer the new theory of n-dimensional linear algebra.

格拉斯曼的代数遵循除外积交换律之外的所有常规定律——和汉密尔顿一样，他知道为他的新代数系统建立这些规则很重要。但它比汉密尔顿的系统更进一步，因为它不仅包含向量，还包含张量代数的萌芽。不过，在这个故事的这个阶段，必须说格拉斯曼自学的数学风格相当令人费解。

Grassmann’s algebra obeyed all the usual laws except the commutative law for outer products—like Hamilton, he knew it was important to establish these rules for his new algebraic system. But it went further than Hamilton’s system in that it contained not just vectors but the germ of tensor algebra. At this point in the story, though, it must be said that Grassmann’s self-taught mathematical style was rather impenetrable.

汉密尔顿在 1853 年出版的 800 页的《四元数讲座》中并没有更清晰地表达。首先，他和格拉斯曼都试图表达他们的同行很难理解的全新思想 — — 矢量分析、线性代数的基本原理等等：汉密尔顿是三维复数和旋转，格拉斯曼是n维矢量空间和原张量代数。他们也都开始发展矢量微积分（下一章会详细介绍）。但是他们工作的原创性意味着他们每个人都必须发明新词来描述他们所做的事情 — — 与最终流行起来的少数几个词（尤其是我刚才提到的“矢量”、“标量”、“内积”和“外积”）相比，这些词更加晦涩难懂，而且最终都是多余的。两人都热衷于哲学思考——汉密尔顿的时间哲学和格拉斯曼对数学基本概念的哲学方法。例如，格拉斯曼是辩证法的忠实拥护者，他运用辩证法将概念分为“相等-不同”、“真实-形式”等类别。

Hamilton’s wasn’t much clearer in his expanded 800-page Lectures on Quaternions, which he published in 1853. For a start, there was the fact that both he and Grassmann were trying to articulate brand-new ideas that their peers would find difficult to digest anyway—the fundamentals of vector analysis, linear algebra, and more: in Hamilton’s case, three-dimensional complex numbers and rotations, and in Grassmann’s, n-dimensional vector spaces and prototensor algebra. They both began to develop vector calculus, too (of which more in the next chapter). But the originality of their work meant that each of them had to invent new words to describe what they were doing—dozens more opaque and ultimately superfluous words than the few that finally caught on (notably the “vector,” “scalar,” “inner,” and “outer” products that I just mentioned). Both men also had a penchant for an excess of philosophising—Hamilton’s philosophy of time, and Grassmann’s philosophical approach to the fundamental concepts of mathematics. For instance, Grassmann was a devotee of dialectics, and he applied it in categorising concepts as “equal-different,” “real-formal,” and so on.

汉密尔顿已经享有数学天才的声誉，而格拉斯曼只是一位默默无闻的乡村教师，这很有帮助——他居然能为《演讲》找到出版商，这真是太神奇了。就连汉密尔顿在出版《演讲》时也遇到了麻烦，他不久后告诉一位朋友：

It helped Hamilton’s case that he already had a reputation for mathematical genius, whereas Grassmann was an unknown country schoolteacher—it’s amazing that he managed to find a publisher at all for Ausdehnungslehre. Even Hamilton had trouble publishing his Lectures, as he told a friend shortly afterward:

它需要一定的科学声誉资本，这些声誉是在过去几年中积累起来的，这样才能使它成为危险的轻率行为。冒险出版一部虽然本质上相当保守，但却充满革命气息的作品。这是我不得不经历的磨难的一部分，是人生战斗中的一个插曲，我知道即使是坦率友好的人也会暗中，或者也可能是公开地谴责或嘲笑我，因为他们认为我的创新是可怕的^。9

It required a certain capital of scientific reputation, amassed in former years, to make it other than dangerously imprudent to hazard the publication of a work which has, although at bottom quite conservative, a highly revolutionary air. It was a part of the ordeal through which I had to pass, an episode in the battle of life, to know that even candid and friendly people secretly, or, as it might happen, openly, censured or ridiculed me, for what appeared to them my monstrous innovations.⁹

他一定对这本书的反响感到失望。即使是约翰·赫歇尔，他在 1847 年听汉密尔顿讲授他的新发现时也非常兴奋，但他也写信给他说，《讲座》“任何人读完都要花十二个月，消化则要花一生的时间。”六年后，当他再次阅读这本书时，他理解得稍微容易一些，但读了三章之后，他绝望地放弃了，因为他对书中分散注意力、独特的哲学感到失望。他建议汉密尔顿，如果他专注于更清楚地解释他的方法的数学算法和术语，那么读者可能会“更好地准备好接受你的形而上学解释。” ¹⁰

He must have been disappointed at the book’s reception. Even John Herschel, who had been so excited when he heard Hamilton lecture on his new discovery back in 1847, wrote to him saying the Lectures would “take any man a twelvemonth to read, and near a lifetime to digest.” He grasped it a little more easily when he came back to it six years later, but after three chapters he gave up in despair over the distracting, idiosyncratic philosophy. He advised Hamilton that if he focussed on explaining his method’s mathematical algorithms and terminology more clearly, then readers might be “better prepared to go along with you in your metaphysical explanations.”¹⁰

读者对格拉斯曼将哲学和数学奇怪地交织在一起感到更加意外，因为他自学数学的事实表明他不熟悉知名数学家使用的数学语言。在互联网出现之前，每当有新论文发表时，格拉斯曼就必须前往柏林寻找学术图书馆——如果他真的听说有新论文需要跟进的话——即使在今天，从什切青到柏林最快的火车路线单程也需要近两个小时。这是另一个例子，说明为什么当时的数学进步往往出现在那些能够培养关系良好的学者群体的国家和机构中。

Readers were even less prepared for Grassmann’s strange interweaving of philosophy and mathematics, because the fact that he was self-taught in maths showed up in his lack of fluency in the mathematical language used by established mathematicians. In those days before the internet, every time a new paper was published Grassmann would have had to travel to Berlin to find an academic library—if indeed he ever heard there was a new paper to follow up—and even today, the fastest train route from Szcezcin to Berlin is almost a two-hour journey each way. It’s another example of why mathematical progress at that time tended to arise in those countries and institutions that were able to foster well-connected communities of scholars.

然而， 《解说》于 1844 年出版后，格拉斯曼立即将这本书的副本寄给了德高望重的高斯，他是当时德国数学界的领军人物，也许是当时世界上最优秀的数学家。高斯的回复相当冷淡，他说他已经研究了同样的想法半个世纪，并在 1831 年发表了其中一些。我提到了他的开创了复数的几何表示，格拉斯曼和汉密尔顿在发现时都不知道这一点，但正如我们将在第 10 章中看到的那样，高斯还开创了现在被称为“微分几何”的学科。格拉斯曼对“内”积和“外”积的研究也为这一分支做出了贡献，但在 1844 年，高斯觉得格拉斯曼的语言丛林中并没有什么新东西。在回复中，高斯说他很忙，需要时间来熟悉格拉斯曼的“特殊”术语——但显然，他一直没有时间去做。

Nevertheless, as soon as Ausdehnungslehre was published in 1844, Grassmann sent a copy to the venerable Gauss, the leading German mathematician and perhaps the best in the world at that time. Gauss replied rather dismissively, saying that he’d been working on the same ideas for half a century and had published some of them in 1831. I mentioned his pioneering geometrical representation of complex numbers, which neither Grassmann nor Hamilton had known of when they made their discoveries, but, as we’ll see in chapter 10, Gauss also initiated what is now known as “differential geometry.” Grassmann’s work on “inner” and “outer” products would also contribute to this branch, but in 1844 it seemed to Gauss that there was nothing new in Grassmann’s dense jungle of words. In his reply Gauss said that he was very busy, and that it would take time to familiarise himself with Grassmann’s “peculiar” terminology—but apparently, he never got around to it.

格拉斯曼没有气馁，他去了莱比锡，会见了高斯以前的学生莫比乌斯，他发表了一些早期的矢量思想。这再次让人想起了 19 世纪中叶的技术状况，因为当时没有电话或电子邮件，无法足不出户进行长距离通信。当然，旅行的好处是可以亲自接触，格拉斯曼的来访是数学“志趣相投的人”的一次欢乐聚会，正如莫比乌斯回忆的那样。格拉斯曼随后写了一封信，羞怯地问他的新朋友是否愿意评论《论数学》，因为他完全有能力评判这部作品的缺点和“书中可能包含的任何优点” ^。11

Undeterred, Grassmann took a trip to Leipzig to meet Möbius, Gauss’s former student, who had published some early vectorial ideas. It’s another reminder of the state of technology in the mid-nineteenth century, for there were no telephones or email to make long-distance communications possible without having to leave home. The upside of having to travel, of course, is the personal contact, and Grassmann’s visit was a convivial meeting of mathematical “kindred spirits,” as Möbius recalled. Grassmann followed it up with a letter diffidently asking his new friend if he would review Ausdehnungslehre, since he would be well able to judge the work for both its weaknesses and “whatever merits the book may contain.”¹¹

等待回复的过程一定令人焦急万分。事实上，可怜的格拉斯曼不得不等了四个月，因为莫比乌斯根本无法理解这样一本创新而又古怪的书。他在回复中说，他曾多次尝试阅读这本书，但每次都无法理解其中的哲学。然而，正如他告诉同事恩斯特·阿佩尔特，当他“强迫”自己浏览这本书时，他觉得格拉斯曼对几何加法和乘法的基本思想（我们今天所称的向量代数）的系统介绍可能对数学的发展有益。因此，正如他在回复中告诉格拉斯曼的那样，他请了一位熟悉哲学的数学家来审阅这本书（他称其为 Drobisch）。但为了以防万一，他建议格拉斯曼最好亲自审阅^。12

It must have been a nail-biting wait for a reply. In fact, poor Grassmann had to wait four months, because Möbius simply couldn’t get his head around such an innovative and eccentric book. In his reply he said he’d made many attempts to read it, but each time he couldn’t get past the philosophy. Still, as he told his colleague Ernst Apelt, when he “forced” himself to skim through it, he felt that Grassmann’s systematic presentation of the fundamental ideas of geometric addition and multiplication—what we know today as the algebra of vectors—had something about it that might be good for the development of mathematics. So, as he told Grassmann in his reply, he’d asked a mathematician familiar with philosophy to review it (someone he referred to as Drobisch). But just in case, he suggested that it might be best for Grassmann to review it himself.¹²

莫比乌斯的哲学同事没有发表评论，格拉斯曼自己也没有勇气发表评论。相反，他将自己的书寄给了《数学与物理档案》的编辑约翰·格鲁纳特。和其他人一样，格鲁纳特也觉得这太难了，但他好心地建议格拉斯曼写一篇自己的评论摘要。于是，格拉斯曼就写了，尽管他的评论并不比这本书本身更有启发性。但至少他成功地将自己的作品介绍给了主流同行——他的评论是《Ausdehnungslehre》收到的唯一一篇评论。阿佩尔特告诉莫比乌斯，它太“奇怪”、太抽象、太不直观了——而阿佩尔特只读过格拉斯曼的评论！他当然不会开始读这本书。在莫比乌斯向海因里希·巴尔策推荐这本书后，他确实试着读了读，但每当他试图进入格拉斯曼的思维过程时，他都会“头晕目眩，眼前一片天蓝”。莫比乌斯回答说，他知道这种感觉！¹³

No review was forthcoming from Möbius’s philosophical colleague, and Grassmann was not so bold as to send out a review himself. Instead, he sent a copy of his book to Johann Grunert, the editor of Archiv der Mathematik und Physik. Like everyone else, Grunert found it too difficult, but he kindly suggested that Grassmann write his own review-summary. And so he did, although it was hardly more enlightening than the tome itself. But at least he’d managed to introduce his work to his mainstream colleagues—his was the only review Ausdehnungslehre received. It was simply too “strange,” too abstract and unintuitive, as Apelt told Möbius— and Apelt had only read Grassmann’s review! He certainly wasn’t about to start on the book. Heinrich Baltzer did try to read it after Möbius recommended it to him, but whenever he tried to enter into Grassmann’s thought-processes, he would “become dizzy and see sky-blue before my eyes.” Möbius replied saying he knew the feeling!¹³

事情就这样继续下去。但格拉斯曼是个斗士。和汉密尔顿一样，他坚信自己的方法将使物理学和几何学中的许多计算变得简单得多，并于 1845 年发表了一篇关于电动力学（即移动带电粒子和改变电磁场（我们现在称之为）的力学）的论文，该论文利用了他的新“外积”。电磁学是一个新颖而令人兴奋的话题，数学家们试图理解越来越多的实验结果——这是汉斯·奥斯特发现电磁学存在的二十五年后，但比詹姆斯·克拉克·麦克斯韦的电磁场理论早了二十年。麦克斯韦的理论将成为矢量历史的转折点，但与此同时，格拉斯曼意识到他的新系统可以更好地描述法国电磁学先驱安德烈-马里·安培于 1826 年发表的一项成果。该成果涉及小电路所施加的磁力的强度和方向——因此，你可以明白为什么矢量或格拉斯曼的“有向线”可能是它的理想语言。但是，当涉及到这种电磁现象的某些无法测试的方面时，格拉斯曼从已发表的实验结果中推导出的公式与安培给出的结果略有不同。细节在这里并不重要——首先，关于安培和格拉斯曼这两个公式中哪一个是更好的实验公式仍然存在争议契合；其次，格拉斯曼的矢量方法与麦克斯韦的方法更相符（关于麦克斯韦的方法我们稍后会详细介绍），因此，在 19 世纪 70 年代，随着麦克斯韦的胜利，格拉斯曼的版本终于获得认可，战胜了安培的版本^。14

And so it went. But Grassmann was a battler. Like Hamilton, he was convinced that his method would make many computations in physics and geometry much simpler, and in 1845 he published a paper on electrodynamics—the mechanics of moving electric particles and changing electromagnetic fields (as we now call them)—which utilised his new “outer product.” Electromagnetism was a new and exciting topic, as mathematicians attempted to make sense of a growing number of experimental results—this was twenty-five years after Hans Øersted discovered the very existence of electromagnetism, but twenty years before James Clerk Maxwell’s theory of the electromagnetic field. Maxwell’s theory will prove a turning point in the story of vectors, but in the meantime, Grassmann realised that his new system could better describe a result published by the French pioneer of electromagnetism André-Marie Ampère, back in 1826. It concerned the strength and direction of the magnetic force exerted by a small electrical circuit—so you can see why vectors or Grassmann’s “directed lines” might be the ideal language for it. But the formula that Grassmann deduced from published experimental results gave some subtly different results from those given by Ampère when it came to certain untestable aspects of this electromagnetic phenomenon. The details do not matter here—except to say, first, that there is still contention over which of the two formulae, Ampère’s or Grassmann’s, is the better experimental fit; and, second, that Grassmann’s vectorial approach fitted better with Maxwell’s, of which more soon, and so Grassmann’s version won out over Ampère’s when it finally achieved recognition in the 1870s, in the wake of Maxwell’s triumph.¹⁴

幸运的是，尽管高斯的数学特质和创新成果不够完美，但这位才华横溢的外行人确实在早期获得了像莫比乌斯这样水平的支持者，莫比乌斯初出茅庐的矢量工作在射影几何和不变量研究中具有重要意义。莫比乌斯还以他非凡的“莫比乌斯带”预示了拓扑学领域的发展：一条两端相连的扭曲纸条，只有一侧，如果你沿着表面画一条线，就会看到这一点。高斯的另一位前学生约翰·林丁也在同一时间发现了这种现象。林丁创造了“拓扑学”一词，如今它指的是研究一般的不变性质，例如表面的边数或物体上的孔数：例如，甜甜圈和茶杯各有一个孔，如果它们都是用橡皮泥或面团制成的，你可以将一个变成另一个，而无需切割或粘合。换句话说，拓扑学并不关注甜甜圈或茶杯的具体形状，而是关注在这种连续的“橡皮泥”变形下不会改变的属性——它将在描述曲面和时空方面发挥作用。至于莫比乌斯，他显然在格拉斯曼的作品中看到了一些特别之处，即使他自己读不了多少，因为他在 1845 年初写信给格拉斯曼，告诉他有一场学术论文竞赛。这正是这位孤独的学校老师所需要的那种同事般的友善——论文主题正合他的胃口。

Luckily, despite his unpolished mathematical idiosyncrasies and innovative results, the gifted young outsider did find early supporters in people of the caliber of Möbius, whose fledgling vectorial work was important in projective geometry and the study of invariants. Möbius also prefigured the field of topology with his remarkable “Möbius band”: a twisted strip of paper joined at the ends that has only one side, as you can see if you trace a line along the surface. Another of Gauss’s former students, Johann Listing, also discovered this phenomenon, around the same time. Listing coined the term “topology,” which today refers to the study of general, invariant properties such as the number of sides of a surface, or the number of holes in an object: for instance, a donut and a teacup each have one hole, and if they were both made of plasticine or dough, you could turn one into the other without cutting or gluing. In other words, topology doesn’t focus on the specific shape of the donut or teacup but on properties that don’t change under this kind of continuous “plasticine” deformation—and it will prove useful in describing curved surfaces and space-times. As for Möbius, he evidently saw something special in Grassmann’s work even if he himself couldn’t read much of it, for he wrote to Grassmann in early 1845 to tell him about an academic essay competition. It was just the sort of collegial kindness the isolated schoolteacher needed—and the essay topic was right up his alley.

在第 3 章中，我提到了牛顿在用大小和方向表达物理量方面的开创性作用，但牛顿的宿敌莱布尼茨也在其中扮演了重要角色。这与符号的力量有关。莱布尼茨说：“值得注意的是，符号有助于发现。这以一种最奇妙的方式减少了思维劳动。”我们在微积分中看到了这一点，莱布尼茨的微分学dy / dx符号比牛顿的ẏ更适合计算和概念理解——约翰·沃利斯在托马斯·哈里奥特的代数本身可以充分利用符号。但莱布尼茨也希望为几何学提供一种类似的省力语言^。15

In chapter 3, I mentioned Newton’s pioneering role in expressing physical quantities in terms of both magnitude and direction, but Newton’s nemesis Leibniz also has a part to play in the story. It has to do with the power of symbols. “It is worth noting,” Leibniz said, “that notation facilitates discovery. This, in a most wonderful way, reduces the mind’s labors.” We’ve seen this in calculus, where the Leibnizian dy/dx notation for differential calculus is better adapted to computations and conceptual understanding than Newton’s ẏ—and John Wallis had seen it in Thomas Harriot’s full use of symbols in algebra itself. But Leibniz wanted a similar labour saving language for geometry, too.¹⁵

他在 1679 年写给荷兰物理学家克里斯蒂安·惠更斯的一封信中表达了这一想法。惠更斯是光的波动理论和许多其他事物（包括摆钟）的先驱。（虽然摆钟在今天看来似乎是一件古雅的遗物，但它是精确计时的一大进步。半个世纪前，在他们进行下落运动实验时，伽利略不得不用滴水的“水钟”来计时，而哈里奥特则用他的脉搏来计时。）莱布尼茨向惠更斯建议，需要一种新的代数形式，一种“明显是几何或线性的代数，能够直接表达情形，就像（普通）代数直接表达量级一样。”（他所说的“情形”是指空间中的相对位置。）笛卡尔率先使用坐标来描述此类位置，但莱布尼茨在寻找某种更清晰的几何形式——事后看来，我们可以将其想象成一种独立于坐标的矢量箭头或格拉斯曼有向线的语言。莱布尼茨本人并没有找到这样的系统，他写给惠更斯的信直到 1833 年才发表——此时莫比乌斯和贝拉维蒂斯已经在矢量方向上迈出了试探性的一步。但 1845 年的征文竞赛旨在挑战数学家更全面地回答莱布尼茨的建议，奖项将于 1846 年颁发，以纪念莱布尼茨 200 岁诞辰。事实证明，挑战是如此巨大，以至于格拉斯曼是唯一一个参赛的人。这是他发表《引言》之后的第一个好运，因为他的获奖论文发表于 1847 年，附录由莫比乌斯撰写，莫比乌斯的名气和人脉关系使其他数学家更有可能注意到他的门生。

He expressed the idea in a 1679 letter to the Dutch physicist Christian Huygens, pioneer of the wave theory of light and many other things— including the pendulum clock. (Though it seems a quaint relic today, the pendulum clock was a great advance in accurate timekeeping. In their experiments on falling motion half a century earlier, Galileo had had to measure time by a dripping “water clock” and Harriot had used his pulse.) Leibniz suggested to Huygens that a new form of algebra was needed, one that is “distinctly geometrical or linear and which will express situation directly as [ordinary] algebra expresses magnitude directly.” (By “situation” he meant relative position in space.) Descartes had spearheaded the use of coordinates to describe such positions, but Leibniz was looking for something more clearly geometrical—something that we, with hindsight, might imagine as a language of vectorial arrows or Grassmannian directed lines, independent of coordinates. Leibniz himself didn’t find such a system, and his letter to Huygens was published only in 1833—by which time Möbius and Bellavitis had already taken tentative steps in the vectorial direction. But the 1845 essay competition was designed to give mathematicians the challenge of answering Leibniz’s suggestion more fully, and the prize was to be awarded in 1846, in honour of Leibniz’s two hundredth birthday. As it turned out, the challenge was so great that Grassmann was the only one who entered. It was his first piece of luck after publishing Ausdehnungslehre, for his prize-winning paper was published in 1847, with an appendix by Möbius, whose fame and connections made it more likely that other mathematicians would take notice of his protégé.

这篇文章的可读性并不比《论证》高，尽管它确实有助于将格拉斯曼的书带入更广泛的数学界。但当汉密尔顿终于在 1850 年左右听说这本书时，它以难度高而闻名，他觉得自己可能得学会抽烟才能读完它！然而，当他在 1852 年秋天开始阅读它时，他告诉德摩根这是“一部非常原创的作品”，“如果德国人认为我值得关注，他们可能会建立在我看来，格拉斯曼的思想与我的思想相竞争，但直到我的观点形成并发表很久之后，我才意识到这一点。”三个月后，他告诉德·摩根，他仍然在“怀着极大的钦佩和兴趣”阅读格拉斯曼的作品，尽管有时他发现用德语阅读很困难。他很自然地感到惊讶，格拉斯曼“以最明显和最完美的独立性”找到了与他的四元数如此相似的系统^。16

Not that the essay was much more readable than Ausdehnungslehre, although it did help to bring Grassmann’s book into the wider mathematical community. But when Hamilton finally heard of it, sometime around 1850, such was its reputation for difficulty that he thought he might have to learn to smoke in order to get through it! When he began to read it in the fall of 1852, however, he told De Morgan it was “a very original work,” which “the Germans, if they think me worth noticing, will perhaps set up in rivalship with mine, but which I did not see till long after my own views were formed and published.” Three months later, he told De Morgan he was still reading Grassmann “with great admiration and interest,” although some days he found it hard to read in German. He was amazed, naturally enough, that Grassmann had, “with the most obvious and perfect independence,” found such a similar system to his quaternions.¹⁶

两天后，他的热情有所减退，但他再次写信给德·摩根，说他认为格拉斯曼是“一位伟大的、最具德国特色的天才”。但他很难理解格拉斯曼所有不同类型的乘法（他有比我们在向量张量代数中学到的两种更多的乘法）。“我认为我确实理解了他的外积，这对于一个没有学会抽烟的人来说已经说明了些什么。甚至他的内积……我也能很好地接受[因为它们]与我的四元数的‘标量部分’非常相似，他的‘外积’与我的‘矢量部分’非常相似。”（正如我所提到的，汉密尔顿的四元数乘积的“标量和矢量部分”是我们矢量的“标量和矢量积”。）¹⁷

Two days later, his enthusiasm had dimmed somewhat, but he wrote to De Morgan again, saying that he considered Grassmann “a great and most German genius.” But he was having trouble grasping all of Grassmann’s different kinds of multiplication (he had many more than the two that have come down to us in vector-tensor algebra). “His outer products I think I do understand, and that is saying something for a person who has not learned to smoke. And even his inner products … I can swallow pretty well [for they] have much analogy to my ‘scalar parts’ of a quaternion, and his ‘outer products’ to my ‘vector parts.’ ” (As I mentioned, Hamilton’s “scalar and vector parts” of a product of quaternions are our “scalar and vector products” of vectors.)¹⁷

德·摩根对汉密尔顿在代数理论方面的进展非常着迷，一周后他又收到了更新。汉密尔顿说，他当时正和女儿一起读书，女儿从他身后看着他，“对哲学家的愚蠢感到好笑”，这时他注意到，虽然他自己是从约翰·沃伦那里得到了添加“线条”的想法，但格拉斯曼似乎是自己想出来的。（他不知道贾斯图斯·格拉斯曼——直到今天也没有人知道贾斯图斯是如何想到几何代数的。）然而，汉密尔顿继续说，格拉斯曼花了 139 页“鸵鸟般的胃”才切中要点——有向线（或矢量）的实际代数。汉密尔顿说，他很早就想到了这一点，并把它作为他在 1848 年关于新系统的公开演讲的起点。大概他认为这样做可以让听众不需要铁打的“鸵鸟”胃来消化这种新数学。尽管如此，这些讲座的出版版本仍然很难读懂，因为汉密尔顿非常小心地解释他的新代数学中的每一个基本步骤，以至于他使用了大量奇怪的术语——包括他所用的所有新词，比如格拉斯曼 (Grassmann) 曾创造过这种代数。但他确实有一种对话式的风格，他还用实际的例子来说明他的新向量代数——特别是来自天文学和他作为爱尔兰皇家天文学家的工作^。18

De Morgan was fascinated by Hamilton’s progress with Ausdehnungslehre, and a week later he received another update. Hamilton said that he’d been reading with his daughter looking over his shoulder, “amused at the folly of philosophers,” when he noticed that while he himself had got the idea of adding “lines” from John Warren, Grassmann seemed to have worked it out himself. (He didn’t know about Justus Grassmann—and even today no one knows how Justus came to his idea of geometrical algebra.) However, Hamilton continued, Grassmann took 139 “ostrich-stomachneeding” pages to get to the point—the actual algebra of directed lines (or vectors). Hamilton had conceived this early, he said, and had taken it as the starting point of his 1848 public lectures on his new system. Presumably he thought he was thereby sparing his listeners the need for a cast-iron “ostrich” stomach to digest this new mathematics. Still, the published version of those lectures was tough going, because Hamilton took so much care to explain each fundamental step in his new algebra that he employed an excess of strange terminology—including all the new words that he, like Grassmann, had coined. But he did have a conversational style, and he also illustrated his new vector algebra with practical examples—particularly from astronomy and his work as Ireland’s royal astronomer.¹⁸

在听说格拉斯曼之前，汉密尔顿就开始将这些讲座扩展成一本书，由此产生的《四元数讲座》在他给德·摩根的一系列信件后不久出版。汉密尔顿使用希腊字母来区分矢量和数字，数字通常用拉丁字母表示——在他的序言第 62 页，在区分“普通代数”（其中ab = ba）和他的新非交换矢量代数（其中 αβ = −βα）时，汉密尔顿承认格拉斯曼提出了他自己的“一种非常新颖和非凡的斜线（或有向线）非交换乘法”。但他明确表示，他自己的工作与“深刻的哲学家”格拉斯曼的工作无关，他指出格拉斯曼无法将二维复平面扩展到整个空间。

Hamilton had begun expanding these lectures into a book before he’d heard of Grassmann, and the resulting Lectures on Quaternions was published not long after his series of letters to De Morgan. Hamilton used Greek letters to distinguish vectors from numbers, which are usually signified by Latin letters—and on page 62 of his preface, when distinguishing “ordinary algebra,” where ab = ba, from his new noncommutative vector algebra, where αβ = −βα, Hamilton acknowledged that Grassmann had come up with his own “species of non-commutative multiplication for inclined [or directed] lines in a very original and remarkable work.” But he made it clear that his own work was independent of the “profound philosopher” Grassmann’s, and he noted that Grassmann had not been able to extend the two-dimensional complex plane to all of space.

这是两次发现之间的区别。汉密尔顿的四元数主要用于处理三维几何运算，如旋转，正如我所提到的，它们已经找到了一些非凡的应用。格拉斯曼的系统更加抽象，这使得它在三维物理的直接应用中不如矢量和四元数有用，但在后来的推广中更有用，最终导致了现代张量分析。

This is the nub of the difference between the two discoveries. Hamilton’s quaternions were designed largely to handle algebraically threedimensional geometrical operations such as rotations, and they have found some extraordinary applications as I’ve mentioned. Grassmann’s system was more abstract, which made it less useful than vectors and quaternions in immediate applications in 3-D physics, but more useful in the later generalisations that would ultimately lead to modern tensor analysis.

当然，汉密尔顿或格拉斯曼当时并没有预见到这一切。其他人也没有。尽管汉密尔顿早年就声名鹊起，但在 19 世纪 50 年代和 60 年代初，甚至连他最初将复数表示为实数的有序对的想法都没有实现，更不用说四元数了。格拉斯曼的想法甚至没有取得任何进展。但他和汉密尔顿从未放弃。我之前提到过汉密尔顿 1859 年写给新同事 (Peter Guthrie Tait) 的一封欣喜若狂的信，信中谈到他们在将四元数应用于物理学方面的进展，“还有什么比这更简单或更令人满意的吗？你不觉得，也不认为，我们走在正确的道路上，以后会得到感谢吗？别在意什么时候……”格拉斯曼也有类似的感受：在 1862 年版的《解说》中——以更数学、更严谨的方式重新编写，更少的哲学风格，加上一些新材料——他在序言的结尾写道：“我完全相信，我在这里介绍的科学上所付出的劳动不会白费，这些劳动占据了我生命的大部分时间，并耗费了我最艰苦的努力。”他希望不要显得傲慢，但他也希望他的想法最终能够结出硕果，无论它们被搁置多久。

Not that Hamilton—or Grassmann—foresaw all this at the time, of course. Neither did anyone else. Despite Hamilton’s earlier fame, in the 1850s and early 1860s not even his initial idea of representing complex numbers as ordered pairs of real numbers had taken off, let alone his quaternions. Grassmann’s ideas had made even less headway. Yet he and Hamilton never gave up. I mentioned earlier Hamilton’s joyous 1859 letter to a new colleague (Peter Guthrie Tait), saying of their progress in applying quaternions to physics, “Could anything be simpler or more satisfactory? Do you not feel, as well as think, that we are on a right track, and shall be thanked hereafter? Never mind when....” Grassmann felt similarly: in the 1862 version of Ausdehnungslehre—a reworking in a more mathematical, less philosophical style, plus some new material—he ended the foreword by saying, “I remain completely confident that the labor which I have expended on the science presented here and which has demanded a significant part of my life as well as the most strenuous application of my powers, will not be lost.” He hoped not to seem arrogant, but he also hoped his ideas would eventually bear fruit, no matter how long they might lie dormant.

他们确实沉寂了下来，因为第二版《Ausdehnungslehre 》的成绩并不比第一版好。

And lie dormant they did, for the second version of Ausdehnungslehre fared little better than the first.

欧几里得几何学非常严谨，在两千多年中一直占据主导地位。问题是，它是一门日常“平面”空间的科学：例如书页、石板、桌面、沙地——任何欧几里得都能在其上画出永不相交的平行线和内角和为 180° 的三角形的表面。但很明显，在球体或地球仪等曲面上，两条平行线确实会相交。看看两条相邻的经线穿过赤道时的样子：它们一开始是平行的，但随着向北移动，它们逐渐靠近，直到在北极相交。因此，当汉密尔顿和格拉斯曼发现可以放弃交换律而采用乘法，同时仍然拥有一致的代数时，其他数学家正在通过放弃欧几里得所谓的平行公设来开发新的“非欧几里得”几何。

Euclidean geometry was beautifully rigorous and had reigned supreme for more than two thousand years. Trouble was, it was a science of everyday “flat” space: the pages of a book, for example, or a slate, a tabletop, a patch of sand—any surface on which Euclid could draw such things as parallel lines that never meet and triangles whose angles add up to 180°. But it was clear that on a curved surface such as a ball or the globe of Earth, two parallel lines do meet. Check out two nearby lines of longitude as they cross the equator: they start off parallel, but as you move north, they gradually move closer together until they meet at the North Pole. So, while Hamilton and Grassmann were discovering that you could ditch the commutative law for multiplication and still have a consistent algebra, other mathematicians were developing new “non-Euclidean” geometries by abandoning Euclid’s so-called parallel postulate.

其中最主要的是德国的高斯、匈牙利的亚诺什·波约伊和俄罗斯的尼古拉·罗巴切夫斯基，他们的发现将这对张量的故事产生了影响——这个故事包括爱因斯坦现在著名的弯曲时空。与此同时，随着非欧几里得几何在 19 世纪 60 年代渗入主流，人们对非交换代数的兴趣也越来越浓厚——也对汉密尔顿将复数作为实数的有序对的代数处理产生了兴趣，而不再是它们在阿尔冈平面上的“欧几里得”几何表示。

Chief among them were Gauss in Germany, Janos Bolyai in Hungary, and Nicolai Lobachevsky in Russia, and their discoveries will have ramifications for the story of tensors—a story that includes Einstein’s nowfamous curved space-time. Meantime, as non-Euclidean geometry filtered into the mainstream in the 1860s, so people became more interested in noncommutative algebras—and also in Hamilton’s algebraic treatment of complex numbers as ordered pairs of real numbers, moving on from their “Euclidean” geometric representation on the Argand plane.

数学和数学家们又花了 60 年的时间才变得足够成熟，不仅发明了张量分析，而且重新审视了“解释学”，并在其抽象性中认识到了新的张量工具。但格拉斯曼等了这么久，在 19 世纪 60 年代和 70 年代，他基本上放弃了数学；他写的几篇论文没有达到他早期的标准，好像他的心已经离开了数学。相反，他把自己巨大的精神能量投入到编写与他的教学相关的科目（德语、拉丁语、数学、音乐、宗教和植物学）的教科书和论文中。和汉密尔顿一样，他也学习了梵语。19 世纪 70 年代，他翻译了 1123 页的《梨俱吠陀》，并出版了自己关于这部印度教经典的更长的译本——这项艰巨的劳动使他当之无愧地获得了美国东方学会会员和 1876 年图宾根大学的荣誉博士学位。他完成了所有这些工作，包括教科书和语言文学，以及大部分数学和物理，同时还从事全职教学并抚养 11 个孩子（其中 4 个已安葬）——他于 1849 年结婚，那时他首次出版了《离题说》五年后。

It would take yet another sixty years for mathematics, and mathematicians, to become sophisticated enough not only to invent tensor analysis, but also to revisit Ausdehnungslehre and to recognise in its very abstractness new tensorial tools. But that was far too long for Grassmann to wait, and in the 1860s and 1870s he turned mostly away from mathematics; the few papers he wrote were not up to his earlier standard, as if his heart had gone out of it. Instead, he put his prodigious mental energy into writing textbooks and papers on subjects related to his teaching—German, Latin, maths, music, religion, and botany. Like Hamilton, he’d also learned Sanskrit, and in the 1870s he made a 1,123-page translation of the Rig Veda and published his own even longer work on the Hindu classic—exhausting labour that deservedly earned him membership in the American Oriental Society and an honorary doctorate from the University of Tübingen, in 1876. He did all this, the textbooks and the philology, as well as much of his maths and physics, while teaching full time and raising eleven children (and burying four of them)—he’d married in 1849, five years after he first published Ausdehnungslehre.

为了便于理解政治背景，我还要补充一点，格拉斯曼除了在 1848-49 年求爱和结婚外，还和兄弟一起出版了一份每周的政治报纸——格拉斯曼希望看到德国在君主立宪制下统一，但当时，奥托·冯·俾斯麦等人反对宪法改革。经过几次领土战争和二十年，新的德意志帝国才得以统一——俾斯麦担任总理。格拉斯曼创办报纸时，爱尔兰最紧迫的政治问题是日益高涨的罗马天主教独立运动——但汉密尔顿是坚定的圣公会教徒（当时爱尔兰的国教，因为他是一位爱尔兰学者（1891年），爱尔兰当时还是英国的一部分。他没有把精力投入政治，而是致力于推动科学，担任爱尔兰皇家科学院院长，并为英国科学促进会做出积极贡献。

For some political context, I should add that as well as courting and marrying in 1848–49, Grassmann had joined his brother in publishing a weekly political newspaper in those years—Grassmann wanted to see Germany unified under a constitutional monarchy, but at the time, the likes of Otto von Bismarck opposed constitutional reform. It would take several territorial wars and two decades before unification of a new German empire was achieved—with Bismarck as chancellor. While Grassmann was producing his newspaper, the most pressing political issue in Ireland was the growing movement for Roman Catholic independence—but Hamilton was a staunch Anglican (the national religion of Ireland at that stage, for the country was still part of the United Kingdom). He put his energy not into politics but into the promotion of science, through his role as president of the Royal Irish Academy, and his active contributions to the British Association for the Advancement of Science.

• • •

• • •

汉密尔顿从未放弃对四元数应用的追求。除了《四元数讲座》之外，他还发表了 100 多篇有关该主题的论文。尽管他声名显赫，数学实力雄厚，但仍有许多人认为，他几乎将生命的最后 20 年全部奉献给了四元数，这是自欺欺人。人们普遍认为，四元数虽然很出色，但用处不大。

Hamilton never gave up his pursuit of quaternion applications. Aside from his Lectures on Quaternions, he published more than a hundred papers on the subject. Despite his fame and mainstream mathematical prowess, there were many who thought he was deluding himself by devoting the last twenty years of his life almost solely to quaternions. There was a widespread feeling that quaternions were brilliant, but they weren’t all that useful.

然而，随着新一代的崛起，以及麦克斯韦和彼得·格思里·泰特进入我们故事的下一章，汉密尔顿的命运开始改变。泰特成为四元数的坚定支持者，其他人正是通过他和麦克斯韦的工作发展了今天大学里教授的现代矢量分析。可怜的格拉斯曼的思想将继续被证明过于笼统——而且很难与他的“奇怪”哲学和“特殊”术语区分开来——无法直接产生影响。我们稍后会再次与他联系，但即便如此，矢量分析和非欧几里得几何最初将我们引向张量^。19

Hamilton’s fortunes began to change, however, with the rise of a new generation, and the entrance of Maxwell and Peter Guthrie Tait in the next chapter of our story. Tait became a fierce supporter of quaternions, and it was largely through his and Maxwell’s work that others would develop the modern vector analysis taught in university today. Poor Grassmann’s ideas would continue to prove too general—and too difficult to disentangle from his “strange” philosophy and his “peculiar” terminology—to be a direct influence. We’ll check in with him again later, but even so, it is vector analysis and non-Euclidean geometry that will initially lead us to tensors.¹⁹

绘制电磁矢量场

Hatching the Electromagnetic Vector Field

尽管赫尔曼·格拉斯曼的作品难以理解且过于抽象，但威廉·罗文·汉密尔顿而非格拉斯曼成为当今教授的矢量分析的直接联系者，这在一定程度上也是历史的偶然。因为彼得·格思里·泰特和詹姆斯·克拉克·麦克斯韦在学生时代相识，无疑是命运的安排。

Although Hermann Grassmann’s work was difficult to penetrate and excessively abstract, it is also partly an accident of history that William Rowan Hamilton, not Grassmann, is the direct link to the vector analysis taught today. For surely it was fate that brought Peter Guthrie Tait and James Clerk Maxwell together as schoolboys.

他们均于 1831 年出生于苏格兰——比汉密尔顿宣布他在四元数问题上取得的突破早 12 年——从十岁起，他们都进入了著名的爱丁堡学院学习。对于没有母亲的马克斯韦尔来说，最初的经历并不愉快。他是一个古怪的孩子，由溺爱他、不循规蹈矩的父亲抚养长大。他的父亲是一名律师，也是格伦莱尔的领主，格伦莱尔是爱丁堡西南约 90 英里的一处家族庄园。因此，当他来到这所大城市的学校时，马克斯韦尔是个怪人，幽默感古怪，说话犹豫不决，穿着舒适而非时尚的自制衣服——因为所有这些违反班级规范的行为，他受到了无情的欺负。他的学习成绩平平无奇：作为一个自主学习者，他最初觉得学校很无聊；再加上他的语速缓慢、性格腼腆，他得到了“傻瓜”的绰号，所以你可以想象，这个绰号是如何让他成为一个“等着”被打的男孩的^。1

They were both born in Scotland in 1831—twelve years before Hamilton announced his quaternion breakthrough—and from the age of ten they both attended the prestigious Edinburgh Academy. Not that it was a happy experience for the motherless young Maxwell at first. He was an eccentric child, being raised by his doting and unconventional father, a barrister and the Laird of Glenlair, the family estate some ninety miles southwest of Edinburgh. So when he arrived at the big city school, Maxwell was an odd figure, with a quirky sense of humour, a hesitant way of speaking, and homemade clothes that were comfortable rather than fashionable— and for all these transgressions of the class code he was mercilessly bullied. It didn’t help that his academic performance was indifferent: as a selfdirected learner he initially found school boring; this, together with his slow speech and shyness, led to the nickname “Dafty,” so you can imagine how that singled him out as a boy “waiting” to be punched.¹

对他和科学界的未来来说幸运的是，他温和的同学刘易斯·坎贝尔与麦克斯韦成为了朋友，最终他得以安定下来，与坎贝尔和泰特一起成为尖子生。很明显，这是一场富有成效的友好竞争，泰特和麦克斯韦很快也成为了好朋友。和麦克斯韦一样，泰特也失去了父母——他的父亲曾是一位公爵的秘书——并从乡下搬到了爱丁堡，尽管他的家乡达尔基斯离这里只有八英里。

Fortunately for him, and for the future of science, his gentle classmate Lewis Campbell befriended Maxwell, so that eventually he was able to settle down and become one of the top students, along with Campbell and Tait. Tellingly, it was a productive and friendly rivalry, and Tait and Maxwell soon became firm friends, too. Like Maxwell, Tait had lost a parent—his father, who’d been secretary to a duke—and had moved to Edinburgh from the country, although his hometown of Dalkeith was only eight miles away.

爱丁堡学院是一所相对进步的学校，泰特、麦克斯韦和坎贝尔在那里表现出色。16 岁时，他们都考上了爱丁堡大学，但一年后坎贝尔转学到牛津大学——他成为了一名古典学家和英国国教牧师，后来成为他童年好友麦克斯韦的一位极具同理心的传记作者——而泰特则转学到剑桥大学，在那里他以 1852 年毕业的高级考官的身份超越了所有同龄人。高级考官是 Tripos 考试中成绩最好的学生，这是一项为期八天的严酷数学考试，包括大约十六场难度递增的考试。数学被视为所有学生的重要心理准备，无论他们的最终职业是什么——如果你能成为得分最高的考官之一，你就一定会有一条良好的职业道路。泰特非常激动，他先是把这个消息电报给家人，然后给他以前的爱丁堡老师写信，“我激动得说不出话来，就像歌里唱的：我是高级牧马人！”他还获得了久负盛名的史密斯奖，该奖是针对更为复杂的数学考试中成绩最优秀的学生颁发的——总的来说，这是一项非常了不起的成就，他和麦克斯韦的母校为他举行了特别的庆祝活动^。2

The Edinburgh Academy was a relatively progressive school, and Tait, Maxwell, and Campbell excelled there. Then, at sixteen, they all went on to Edinburgh University, although after a year Campbell transferred to Oxford—he became a classicist and an Anglican minister, and later a wonderfully empathetic biographer of his childhood friend Maxwell—while Tait transferred to Cambridge, where he outshone all his peers, graduating as the Senior Wrangler of 1852. The Senior Wrangler was the student with the best marks in the Tripos, a grueling series of around sixteen increasingly difficult mathematics examinations, held over eight days. Mathematics was seen as important mental preparation for all students no matter their ultimate profession—and if you managed to be one of the highest scoring wranglers you were assured of a good career pathway. Tait was so excited that first he telegraphed the news to his family, and then he wrote to his erstwhile Edinburgh teachers, “I’m all in a flutter, I scarcely can utter, etc, as the song has it: i am senior wrangler!” He also took out the prestigious Smith’s Prize, for the top student in an even more complex series of maths exams—and all up it was such a fine achievement that his and Maxwell’s old school held a special celebration in his honour.²

毕业后，泰特获得了剑桥大学的奖学金——这是成为顶尖论文写作者的好处之一，因为这让他获得了一份工作和教学经验，这很可能让他以后在剑桥大学或其他一流大学获得永久讲师职位。不过，目前，他沉浸在新获得的安全感中，没有了备考的苦差事。他甚至给自己买了一本刚刚出版的汉密尔顿的《四元数讲座》 ——他不知道这本书讲的是什么，但他对“四元数”这个奇怪的名字很感兴趣。他在一次旅行中读了前六章暑假期间，麦克斯韦参加了一次射击之旅，但回到剑桥后，他的教学和研究占据了他所有的时间，他把四元数放在了一边。（相比之下，麦克斯韦拒绝打猎或钓鱼，因为他热爱动物。他特别擅长与马打交道，身边总是有一只心爱的狗。）

After graduating, Tait won a fellowship at Cambridge—one of the benefits of being a top wrangler, for it gave him a job and teaching experience that would presumably lead to a permanent lectureship later, at Cambridge or some other leading institution. For the moment, though, he reveled in his newfound security, free of the grind of exam preparation. He even treated himself to a copy of Hamilton’s Lectures on Quaternions, hot off the press—he had no idea what it was about, but he was intrigued by the strange name “quaternion.” He read the first six chapters while away on a summer holiday shooting trip, but once back at Cambridge, his teaching and research took up all his time, and he laid Quaternions aside. (Maxwell, by contrast, refused to hunt or fish, such was his love of animals. He was especially brilliant with horses, and he always had a much-loved dog by his side.)

泰特在剑桥大学就读期间，麦克斯韦留在了爱丁堡大学——显然是为了取悦他的父亲——但那里优秀而广泛的教学为他提供了完美的训练场，让他能够运用实践和哲学技能，很快他便凭借这些技能改变了物理学。1850 年，在爱丁堡大学获得学位后，他进入剑桥大学继续深造。虽然花了一段时间才适应，但腼腆、笨拙的年轻人达夫蒂很快结识了“一群”新朋友，因为剑桥大学的人都觉得他“和蔼可亲、风趣幽默”，能够以诙谐、博学的方式谈论任何话题。一位新认识的人在交谈后写道：“我从未见过像他这样的人。”另一位后来回忆道：“在三一学院（麦克斯韦在剑桥大学的学院）认识他的每个人都能回忆起他的一些善举或一些行为，这些都给他留下了不可磨灭的善良印象。” ³

While Tait was at Cambridge, Maxwell had stayed on at Edinburgh— apparently to please his father—but the excellent, broad-ranging teaching there proved the perfect training-ground for the practical and philosophical skill with which he would soon transform physics. Then, after finishing his Edinburgh degree in 1850, he enrolled at Cambridge to further his scientific studies. It took a while to settle in, but soon the shy, awkward young Dafty made “a troop” of new friends, for the Cambridge crowd found him “genial and amusing,” able to converse in a witty and erudite way on any subject at all. “I never met a man like him,” one new acquaintance recorded after such a conversation, while another later reminisced, “Everyone who knew him at Trinity [Maxwell’s college at Cambridge] can recall some kindness or some act of his which has left an ineffaceable impression of his goodness.”³

虽然麦克斯韦对自己的新社交生活感到高兴，但他对当时的剑桥课程并不满意，因为该课程强调技术熟练度而不是深度思考。年底，当他准备考试时，他在一首长诗《牧马人的幻象》中表达了自己的沮丧。以下是他的幽默和他对考试制度的看法：

While Maxwell delighted in his new social life, he was not so impressed with the Cambridge curriculum of the time, with its emphasis on technical proficiency rather than deep thinking. At the end of the year as he was preparing for exams, he expressed his frustration in a long poem, “A Vision of a Wrangler.” Here’s a taste of his humour, and his view of the exam system:

壁炉里闪烁的余烬

这表明十一月是多么的沉闷

雾气笼罩了我麻木的肢体

就像一只拔了毛又瘦的鹅。

当我准备睡觉的时候，我

我颤抖着声音问自己，

如果我读了所有的东西，我

曾经有过丝毫使用^。4

In the grate the flickering embers

Served to show how dull November’s

Fogs had stamped my torpid members

Like a plucked and skinny goose.

And as I prepared for bed, I

Asked myself with voice unsteady,

If of all the stuff I read, I

Ever made the slightest use.⁴

这只是 24 首越来越博学的诗中的第二首，这些诗围绕着他邪恶的新母校的奇幻化身展开，他表达了对一个要求死记硬背的学习制度的愤慨，这个制度要求学生死记硬背，然后给最优秀的学生提供舒适的学术工作——这些争论者随后成为维持这个制度的精英阶层的一部分。这首诗以一个严肃的精神宣言结束，即最好是默默地思考“创造的荣耀”，而不是漫无目的地研究物理学家用来表示自然的超然的数学符号。他写这首非凡的诗时才 21 岁；然而，在 12 年后，他将展示自然的符号表示有多么奇妙。后来，他自己也成为了一名考官和主持人，他将在使 Tripos 问题（以及剑桥教学大纲）与正在发现的令人兴奋的新科学更加相关方面发挥主导作用^。5

This is but the second of twenty-four increasingly erudite verses centred on a fantastical incarnation of his evil new alma mater, as he expresses his outrage at a system that demands slavish rote study and then rewards its best students with cushy academic jobs—these wranglers then becoming part of the elite who keep the system going. The poem ends with a serious spiritual declaration, that it is better to silently contemplate the “glories of Creation” than to mindlessly study the detached mathematical symbols that physicists use to represent nature. He was twenty-one when he wrote this extraordinary poem; in twelve years’ time, however, he would show just how wondrous the symbolic representation of nature can be. Later, he would also become an examiner and moderator himself, and he would play a leading role in making the Tripos questions—and therefore the Cambridge syllabus—much more relevant to the exciting new science being discovered.⁵

然而，1853 年 11 月，随着期末考试的临近，他又写了一长诗——《在十一月激情熄灭后读数学是不明智的信念下写下的诗句！》在诗中，他谈到了年轻人承受的沉重考试压力，以及将学术奖项和学位误认为智慧的愚蠢行为。因此，他更喜欢学习自己感兴趣的东西，而不是死记硬背——泰特对麦克斯韦在大学荣誉学位考试中准备不足感到惊讶。尽管如此——多亏了麦克斯韦“纯粹的智力”，泰特说道——他还是以第二名的成绩毕业，与高级名次 (EJ Routh) 并列史密斯奖。他的父亲在一封感人至深的支持信中表达了自己的骄傲：“[你的表弟乔治] 凌晨 2 点来到我的房间……看了特快列车送来的《星期六泰晤士报》 ，我在早餐前收到了你的来信……虽然你在冠军 [史密斯] 的比赛中与高级牧马人不相上下，但你只比他落后一点点^。 ”至于麦克斯韦的前学校数学老师，他“欣喜若狂”，就像他为泰特的成功而兴奋一样。6

In November 1853, however, as his final exams drew near, he wrote another long poem—“Lines Written under the Conviction That It Is Not Wise to Read Mathematics in November after One’s Fire Is Out!”—in which he speaks of the soul-crushing exam pressure on young people, and the folly of mistaking academic prizes and degrees for wisdom. So mostly he preferred to study what interested him instead of cramming—Tait was amazed at how unprepared Maxwell was for the Tripos. Still—thanks to Maxwell’s “sheer strength of intellect,” as Tait put it—he managed to graduate as Second Wrangler, tying with the Senior Wrangler (E. J. Routh) for the Smith’s Prize. His father expressed his pride in a touchingly supportive letter: “[Your cousin George] came into my room at 2 am … having seen the Saturday Times, received by express train, and I got your letter before breakfast.... As you are equal to the Senior Wrangler in the champion [Smith’s] trial, you are but a very little behind him.” As for Maxwell’s former school maths teacher, he was “beside himself ” with excitement, just as he’d been with Tait’s success.⁶

你可以从 1854 年 2 月麦克斯韦参加的试卷的第 8 个问题中感受到史密斯奖考试的难度：它要求证明现在被称为“斯托克斯定理”的定理——爱尔兰出生的过去几年，数学家乔治·斯托克斯一直在编写史密斯奖论文。1854 年的论文是该定理首次以印刷形式出现——如今，它出现在每本本科微积分教科书中，它将“线积分”与“曲面积分”联系起来。如今，许多学生发现它很难应用，更不用说证明。没有人知道麦克斯韦或他的任何同学是否也证明了这一点，但我想没有，因为麦克斯韦后来将完整的证明归功于威廉·汤姆森，即未来的开尔文勋爵^。7

You can get a feel for the difficulty of the Smith’s Prize exams from question 8 on the February 1854 paper, which Maxwell sat: it asked for a proof of what is now known as “Stokes’s theorem”—the Irish-born mathematician George Stokes had been setting the Smith’s Prize papers for the past few years. The 1854 paper was the first time this theorem appeared in print—today it’s in every undergrad calculus textbook, and it relates a “line integral” to a “surface integral.” Many students today find it difficult enough to apply, let alone prove. No one knows if Maxwell, or any of his fellow examinees, proved it, either, but I suspect not, for Maxwell later credited the full proof to William Thomson, the future Lord Kelvin.⁷

线积分和曲面积分将在我们的故事中扮演重要角色，所以如果你以前没有遇到过它们，让我给你介绍一下它们是什么。我们在学校学到的普通积分学是对独立（水平）变量（通常是x或t）求函数积分——所以当你求积分时，你是在把曲线下所有细长矩形的面积相加，就像我们在图 2.1中看到的那样。相比之下，一个简单的“线积分”例子是图 2.3b，其中我对一条线的一小段ds（在这种情况下是圆）求积分以求出圆周长。换句话说，在“线积分”中，你要把无穷小数量的小长度而不是面积相加——所以它是艾哈迈斯和阿基米德数千年前所做的圆周近似值的精确版本。“曲面积分”对整个曲面做类似的事情——但它需要“二重积分”这样才能在二维中进行积分。例如，在图 2.3a中，我可以使用曲面积分更轻松地找到圆的面积，如您在尾注中看到的那样。8^但这里的关键点是，您不仅可以对水平轴进行积分，还可以沿曲线和表面进行积分——如果使用三重积分，还可以对体积进行积分。

Line and surface integrals will play a role in our story, so in case you haven’t met them before, let me give you an idea of what they are. The ordinary integral calculus we learn at school integrates a function with respect to the independent (horizontal) variable, often x or t—so when you integrate, you’re adding up the areas of all those skinny rectangles under the curve, as we saw in figure 2.1. By contrast, a simple example of a “line integral” is figure 2.3b, where I integrated with respect to a small segment ds of a line—a circle in that case—to find the circumference. In other words, in a “line integral” you’re adding up an infinitesimal number of little lengths rather than areas—so it’s an exact version of the circumference approximations Ahmes and Archimedes made thousands of years ago. A “surface integral” does a similar thing over a whole surface—but it needs a “double integral” so that you can integrate in two dimensions. For example, in figure 2.3a I could have found the area of the circle more easily using a surface integral, as you can see in the endnote.⁸ But the crucial point here is the idea that you can integrate not just with respect to the horizontal axis, but also along curved lines and across surfaces—and through volumes, too, if you use a triple integral.

至于麦克斯韦，十年后，他将充分利用他在史密斯奖考试中第一次遇到的定理，但与此同时，尽管他的考试成绩优异，他却没有获得剑桥奖学金。显然，当权者认为他太粗心了——他公开承认了数学上的缺陷，我相信我们很多人都能感同身受。因此，才华横溢但有时草率的麦克斯韦毕业了，却没有工作，而几个月后，泰特被提升为贝尔法斯特皇后学院的数学教授；他的《四元数》 仍未完成，等待他抽出时间去研究。这表明麦克斯韦的困境也有好的一面，因为他的父亲生活宽裕：在剑桥担任私人教师和在新成立的工人学院做志愿者的间隙，他有时间在家乡格伦莱尔静静地思考迅速发展的电磁学这一新科学——这是一座美丽的庄园，至今仍与麦克斯韦时代的样子一模一样（这在很大程度上要归功于其现任主人邓肯·弗格森^{9 的}努力）。

As for Maxwell, in ten years’ time he would make good use of the theorem he first met in the Smith’s Prize exam, but in the meantime, and despite his outstanding exam results, he wasn’t awarded a Cambridge fellowship. Apparently, the powers-that-be thought he was too careless—a mathematical failing he openly acknowledged, and one I’m sure many of us can identify with. So, the brilliant if sometimes slapdash Maxwell graduated without a job, while a few months later Tait was promoted to professor of mathematics at Queens College, Belfast; his copy of Quaternions still lay unfinished, waiting for him to find the time to study it. Which suggests there was an upside to Maxwell’s predicament, given that his father was comfortably off: between stints as a private tutor at Cambridge and volunteering at the newly founded Working Men’s College, he had time to contemplate the rapidly developing new science of electromagnetism, at home in the peace of Glenlair—a beautiful estate that still looks much as it did in Maxwell’s time (thanks largely to the efforts of its present owner, Duncan Ferguson⁹).

重获自由后，麦克斯韦做的第一件事就是给威廉·汤姆森写信，汤姆森在电学数学方面的工作已经给他留下了深刻印象。麦克斯韦说，他想尽快了解电磁研究的现状，并询问汤姆森能否建议他从哪里开始。

One of the first things Maxwell did in his newfound freedom was write to William Thomson, whose work on the maths of electricity had already made an impression on him. Maxwell said that he wanted to get himself up to speed on the state of electromagnetic research, and he asked if Thomson would advise him on where he should start.

这种大胆的做法并非完全出乎意料，因为他和汤姆森几年前就见过面。那是在 1850 年英国协会在爱丁堡举行的一次会议上，当时 19 岁的麦克斯韦——他仍然很害羞和笨拙，说话带有地方口音，说话犹豫不决——在提问时间勇敢地站起来，就刚刚提交给会议的一篇关于光学和视觉的论文发表了看法。（麦克斯韦也是视觉和色彩研究的先驱，他后来研究如何拍摄出有史以来第一张日常物品的彩色照片——一条格子缎带，他是一位自豪的苏格兰人。）据一位与会者说，观众“半是困惑，半是焦虑，也许还有点怀疑……他们注视着这个看起来粗鲁的年轻人，他用断断续续的口音向他们讲话。”因此，你可以想象，当会议结束后，数学物理学界的一颗新星——26 岁的汤姆森找到麦克斯韦，想进一步了解他的想法时，麦克斯韦是多么惊讶和欣慰^。10

This bold approach was not entirely out of the blue, for he and Thomson had met several years earlier. It was at an Edinburgh meeting of the British Association in 1850, where nineteen-year-old Maxwell—still shy and awkward, with his regional accent and hesitant way of speaking—had bravely stood up to make a point at question time, about a paper on optics and vision that had just been presented to the meeting. (Maxwell was also a pioneer of the study of vision and colour, and he would work out how to create the first-ever colour photograph of an everyday object—a tartan ribbon, proud Scot that he was.) According to one attendee, the audience was “half-puzzled, half-anxious and perhaps somewhat incredulous … as they gazed on the raw-looking young man who, in broken accents, was addressing them.” So, you can imagine Maxwell’s surprise and relief when, after the meeting had finished, he was approached by a rising star in the world of mathematical physics: twenty-six-year-old Thomson, who wanted to know more about his ideas.¹⁰

1854 年，汤姆森慷慨地回复了麦克斯韦的信，在他的建议下，麦克斯韦开始仔细阅读迄今为止已知的所有电磁结果。他本可以在大学学习一些相关的数学和物理学，但在接下来的几页中，我将带你简要回顾一下他需要了解的关键历史知识。在改造这个主题之前，麦克斯韦必须先了解所有可能，从而在 Tait 的帮助下开创了一种全新的物理学方法：矢量场。或者更确切地说，麦克斯韦采用了其他人直观地瞥见的方法，并为其注入了新的活力，使它真正成为一种全新的方法。

Now, in 1854, Thomson replied generously to Maxwell’s letter, and with his advice, Maxwell set himself the task of reading up carefully on all the electromagnetic results known so far. He would have studied some of the relevant maths and physics at university, but in the next few pages I’ll take you on a brief historical journey through the key things he needed to know before he could transform the subject—and thereby inaugurate, with Tait’s help, a brand-new approach to physics: the vector field. Or rather, Maxwell would take an approach that others had intuitively glimpsed and breathe such new life into it that it was, indeed, brand-new.

因此，电力的第一批数学研究是关于静电荷的影响——电荷倾向于停留在原地而不是在电流中自由移动。特别是，1785 年，查尔斯·奥古斯丁·库仑用扭力秤做了他著名的实验，并表明带电粒子的电力与离源距离的平方成反比——就像牛顿引力定律中静止物体的质量施加的力一样。令人惊讶的巧合是，两个如此不同的现象的数学定律竟然具有相同的形式^。11

So, the first mathematical studies of electricity were about the effects of static charges—charges that tend to stay put rather than freely moving in a current. In particular, in 1785 Charles Augustin Coulomb did his famous experiment with a torsion balance and showed that the electric force from a charged particle drops off in inverse-square proportion to the distance from the source—just like Newton’s law of gravity for the force exerted by a stationary object’s mass. It was an astonishing coincidence that the mathematical laws of two such different phenomena should have the same form.¹¹

大约在这个时候，约瑟夫-路易斯·拉格朗日将他强大的数学头脑运用到引力理论的新公式中——就像平方反比定律一样，它很快也被用于描述电学。拉格朗日的公式涉及数学“势”的概念——一个与“势能”和力所做的“功”有关的概念。例如，要克服重力举起重物需要努力——做功，但一旦重物被举起来，如果重物掉下来，就有可能消耗能量做更多的功。你可以在起重机上吊起的巨大爆破球或驱动涡轮机的落水中看到这种应用。然而，要举起爆破球，你需要对它施加力，就像牛顿第二运动定律一样；这种力所做的“功”定义为“力乘以物体移动的距离”。在尾注中，我给出了“功”的数学原理以及它如何得出势能——但主要观点是，功的定义涉及积分^。12

At around this time, Joseph-Louis Lagrange brought his powerful mathematical mind to bear on a new formulation of the theory of gravity—and just like the inverse square law, it would soon prove useful in describing electricity, too. Lagrange’s formulation involved the concept of a mathematical “potential”—an idea related to “potential energy” and the “work” done by a force. For instance, it takes effort—work—to lift a weight against gravity, but once it’s up there it has the potential to expend energy doing more work if it falls. You can see this kind of application in the huge demolition balls hoisted on cranes or in the falling water that drives a turbine. To lift up the demolition ball, though, you need to apply a force to it, à la Newton’s second law of motion; the “work” done by this force is defined as “force times the distance the object is moved.” In the endnote, I’ve given the maths of “work” and how it leads to the potential—but the main point is that the definition of work involves an integral.¹²

如今，“电压”（以伏特命名）可能更适合用来描述电势，它的作用类似于热量中的温度和流体中的压力。如果一根装满水的管道两端存在压力差，那么水就会从高压端流出端到低压端。同样，热量从较热的区域流向较冷的区域。当事物发生这样的变化时，导数就会发挥作用，因此，您不会惊讶地发现，如果V是与力相关的势，那么力的分量是：

Today, “voltage” (named in honour of Volta) might be a more familiar term for electric potential, and its role is similar to that of temperature in heat and pressure in fluids. If there’s a pressure difference between the ends of a pipe filled with water, then the water flows from the high-pressure end to the low-pressure end. Similarly heat flows from a hotter region to a colder one. When things change like this, derivatives come into play, so you won’t be surprised to find that if V is the potential associated with a force, then the force’s components are:

$\frac{\partial 五}{\partial 十}，\frac{\partial 五}{\partial 是}，\frac{\partial 五}{\partial 是}$。

$\frac{\partial V}{\partial x},\frac{\partial V}{\partial y},\frac{\partial V}{\partial z}$.

大写的 d 表示“偏”导数，表示V在整个空间中变化，而不仅仅是像我们熟悉的$\frac{d是}{d十}$. ¹³

The curly d’s indicate “partial” derivatives, which show that V is changing all through space, not just in one direction as with the familiar $\frac{dy}{dx}$.¹³

拉格朗日思想在电学和磁学中的应用是几位数学家几十年工作的成果，首先是皮埃尔-西蒙·拉普拉斯，他比拉格朗日更充分地发展了势理论。然后是无处不在、多才多艺的卡尔·弗里德里希·高斯，了不起的自学成才的数学家乔治·格林（当时他的日常工作是当磨坊主，尽管十年后，43 岁的他以第四名的成绩从剑桥大学毕业）以及拉普拉斯和拉格朗日的学生西蒙-丹尼斯·泊松。1817 年，玛丽·萨默维尔和她的丈夫访问巴黎时，拉普拉斯邀请他们到他的乡间别墅做客——她是英国为数不多的研究并理解他的《天体力学》的人之一。她发现拉普拉斯非常善良、细心，而泊松是一位有趣而活泼的晚宴客人。

The application of Lagrange’s idea to electricity—and to magnetism, too—was the result of the work of several mathematicians over several decades, beginning with Pierre-Simon Laplace, who developed potential theory more fully than Lagrange. Then came the ubiquitous and versatile Carl Friedrich Gauss, the remarkable self-taught mathematician George Green—whose day job at the time was being a miller, although a decade later, at the age of forty-three, he graduated from Cambridge as Fourth Wrangler—and Laplace and Lagrange’s student Siméon-Dénis Poisson. When Mary Somerville and her husband visited Paris in 1817, Laplace invited them to his country house—she was one of the few in Britain who had studied and understood his Mécanique Celeste. She found Laplace very kind and attentive, and Poisson an entertaining and vivacious dinner guest.

然而，索菲·热尔曼与泊松的经历却截然不同，因为她竟敢与他在同一领域竞争。她以匿名形式在法国科学院主办的征文比赛中提交了一篇关于振动表面理论的开创性论文——这是一个通过物理发现（当沙子散落在因声音而振动的盘子上时形成的图案）为数学开辟的全新主题。这是一个非常困难的问题，就像格拉斯曼参加莱布尼茨百年纪念竞赛一样，热尔曼的论文是唯一提交的论文。她的身份很​​可能是一个公开的秘密——但她的自学能力确实显露出来，她的推导并不完全正确，所以她的初稿被拒绝了。她重新修改了论文并再次提交——只有标准足够高才会获奖；除此之外，竞赛仍未结束。也许是因为她是局外人，作为评委之一的泊松表现得相当糟糕，严厉批评了她的第二篇论文（其他评委给了它荣誉奖），然后拿起结论自己研究。她没有气馁，继续改进论文，最终在 1816 年第三次尝试时赢得了奖项。我在这里提到它，不仅仅是为了赞扬杰曼和她不顾女性在科学领域面临的障碍而坚持不懈，也因为振动表面的数学包含一些与大约同一时期为重力、电力和磁力开发的方程和表达式类似的方程和表达式。

Sophie Germain, however, had a rather different experience with Poisson, for she had the temerity to compete in the same playing field as he. She anonymously entered a pioneering paper on the theory of vibrating surfaces—a brand-new topic opened up to mathematics through a physical discovery (the patterns made when sand is scattered on a plate set vibrating by a sound)—in an essay competition sponsored by the French Academy of Sciences. It was such a difficult problem that, just as with Grassmann’s entry in the Leibniz centenary competition, Germain’s was the only essay submitted. It is likely her identity was an open secret—but it’s true that her self-education showed, and her derivations weren’t completely correct, and so her first effort was rejected. She reworked her paper and submitted it again—prizes were only awarded if the standard was high enough; otherwise, the competition remained open. Perhaps because she was an outsider, Poisson, who was one of the judges, behaved rather badly, by severely criticising her second essay (the other judges gave it an honourable mention) and then lifting its conclusion and working on it himself. Undeterred, she kept on improving her paper and finally won the prize with her third attempt, in 1816. I mention it here not just as a shout out to Germain and her persistence despite the obstacles for women in science, but also because the maths of vibrating surfaces includes some similar equations and expressions to those being developed for gravity, electricity, and magnetism around the same time.

例如，拉普拉斯已经证明势能V遵循一个非常简单的方程：

For instance, Laplace had shown that the potential V obeyed a really neat equation:

$\frac{{\partial }^2五}{\partial {十}^2}+\frac{{\partial }^2五}{\partial {是}^2}+\frac{{\partial }^2五}{\partial {是}^2}=0.$

$\frac{{\partial }^{2}V}{\partial x^2}+\frac{{\partial }^{2}V}{\partial y^2}+\frac{{\partial }^{2}V}{\partial z^2}=0.$

泊松后来证明，在物理应用中，右边并不总是零——这取决于你测量的内容和位置。无论哪种情况，麦克斯韦都将左边表达式的物理意义描述为势的“浓度”，在数学上，它今天被称为V的“拉普拉斯算子” ，其中拉普拉斯算子是“微分算子”

Poisson later showed that in physical applications the right-hand side is not always zero—it depends on what and where you’re measuring. Either way, Maxwell described the physical meaning of the expression on the lefthand side as the “concentration” of the potential, and mathematically it is known today as the “Laplacian” of V, where the Laplacian is the “differential operator”

$\frac{{\partial }^2}{\partial {十}^2}+\frac{{\partial }^2}{\partial {是}^2}+\frac{{\partial }^2}{\partial {是}^2}$

$\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}$

（或者麦克斯韦所说的“拉普拉斯算子”）。

(or “Laplace’s operator,” as Maxwell called it).

正如我在第二章中提到的$\frac{d}{d十}$中，“算符”一词仅表示表达式指示您“操作”尚未插入的某个函数（在本例中为求其二阶（偏）导数）。在物理学中，您要操作的函数具有物理意义，例如势能，但从数学上讲，任何适当可微的函数都可以。当我们讨论张量时，函数“插入”时“操作”的思想将变得尤为重要。

As I mentioned in chapter 2 in connection with $\frac{d}{dx}$, the term “operator” just means that the expression directs you to “operate on”—in this case, take the second (partial) derivatives of—some function that is yet to be inserted. In physics, the functions you want to operate on have a physical meaning, such as the potential, but mathematically speaking, any suitably differentiable function will do. The idea of something that “operates” when a function is “inserted” will become especially important when we come to tensors.

与此同时，自然与数学似乎以一种奇妙的方式交织在一起，拉普拉斯算子不仅出现在势理论中，还出现在热、振动和波的理论中：除了使用势V，还可以取温度T的拉普拉斯算子，例如在“热方程”中，或者取波的拉伸距离u，例如在“波动方程”中。然而，这对我们的故事的意义在于，拉格朗日、拉普拉斯和其他先驱者只是使用了笛卡尔坐标和分量。他们有牛顿的矢量概念，但全矢量微积分尚未发明。

Meantime, in the wonderful way that nature and mathematics seem intertwined, the Laplacian appears not only in potential theory but in the theory of heat, vibrations, and waves: instead of using the potential V, you can take the Laplacian of the temperature T, say, as in the “heat equation,” or the stretching distance u in a wave, as in the “wave equation.” The significance of this for our story, though, is that Lagrange, Laplace, and the other pioneers simply used Cartesian coordinates and components. They had the Newtonian idea of a vector, but whole-vector calculus had yet to be invented.

然而，一旦 Tait 静下心来阅读 Hamilton 的四元数，他就会开始以全矢量形式应用势能和拉普拉斯算子——然后 Maxwell 会利用这些矢量量获得前所未有的优势。但那还是 15 年后的事。1854 年，Maxwell 仍在阅读他进入拳击场之前的内容，而 Tait 仍在忙于他在贝尔法斯特的新学术工作。

Once Tait settled down to reading Hamilton’s Quaternions, however, he would begin to apply the potential and Laplacian in full vector form— and Maxwell would then use these vector quantities to unprecedented advantage. But that was still fifteen years in the future. In 1854, Maxwell was still reading up on what came before he entered the ring, and Tait was still busy with his new academic duties in Belfast.

• • •

• • •

好像所有这些新的静电计算还不够具有挑战性，当奥斯特发现电磁学的存在（涉及在流动的电流中移动带电粒子）时，事情变得更加复杂。

As if all this new calculus of static electricity wasn’t challenging enough, when Øersted discovered the existence of electromagnetism—which involved moving charged particles in a flowing electric current—things became a whole lot more complicated.

1821 年，安德烈-马里·安培利用 Øersted 的奇特发现开创了电报技术——他还开始尝试通过实验量化电磁效应，然后用数学表达它们。他取得了惊人的进步，独自开创了电磁学（或他称之为“电动力学”）的数学研究。麦克斯韦称他为“电学界的牛顿”，因为他是一位伟大的实验家和数学理论家。特别是，他使用线积分和面积积分来量化电流和它在各种实验装置中产生的磁力之间的关系——但当他于 1836 年去世时，他仍然没有找到对电磁学的完整通用描述^。14

In 1821, André-Marie Ampère used Øersted’s strange discovery to pioneer telegraphy—and he also began trying to quantify electromagnetic effects experimentally and then express them mathematically. He made amazing progress, single-handedly initiating the mathematical study of electromagnetism (or as he called it, “electrodynamics”). Maxwell dubbed him the “Newton of electricity,” because he was such a great experimenter and mathematical theorist. In particular, he used line and surface integrals to quantify the relationship between electric current and the magnetic force it produces in various experimental set-ups—but when he died in 1836, he still hadn’t managed to find a completely general description of electromagnetism.¹⁴

在安培开始电磁研究的同时，英国人迈克尔·法拉第利用奥斯特的发现开发了一种电动机的原型。然后，在 1831 年，也就是麦克斯韦和泰特出生的那一年，法拉第发现了与 Øersted 的相反的效应：他证明了可以通过移动磁铁穿过线圈来产生电流。（正是相对运动产生了电流，所以你也可以转动线圈而不是磁铁，我们很快就会在图 6.2中看到这种装置。）几个月后，美国人约瑟夫·亨利 (Joseph Henry) 发现了同样的现象，这又是一个科学上独立共同发现的例子。

At the same time as Ampère began his electromagnetic research, Englishman Michael Faraday used Øersted’s discovery to develop a rudimentary prototype of an electric motor. Then, in 1831, the year Maxwell and Tait were born, Faraday discovered the inverse effect to Øersted’s: he showed that you could generate an electric current by moving a magnet through a coil of wire. (It is the relative motion that induces the current, so you can just as well turn the coil instead of the magnet, and we’ll see this set-up shortly, in fig. 6.2.) In yet another example of independent codiscovery in science, the American Joseph Henry discovered the same thing, some months later.

法拉第用一台原型发电机展示了这一非凡现象，后来其他人将其发展成为商业产品。剩下的就成了历史，无论好坏，因为到了 19 世纪后期，世界开始通电。例如，在 19 世纪 80 年代，汤姆森建造了一座房子，这是世界上最早使用电灯的房子之一；而在德国和意大利，爱因斯坦的父亲和叔叔经营着一家创新型科技公司，制造和安装发电机，首次将电力带入企业、公共建筑、街道甚至家庭。当然，今天我们很遗憾大多数大型发电机都是燃煤的——在没有水流或风等自然力的情况下，需要热量来产生蒸汽，才能转动产生电流的巨大线圈。尽管如此，我们中很少有人愿意没有电，所以我们不仅要感谢设计出可持续、气候安全的电力生产方式的现代科学家和工程师，还要感谢试图弄清楚电力如何运作的十九世纪实验学家和数学家。

Faraday demonstrated this remarkable phenomenon with a prototype generator, which others would later develop into commercial propositions. The rest is history, for better and worse, for by the late 1800s the world began to go electric. In the 1880s, for example, Thomson built a house that was one of the first in the world to include electric light, while in Germany and Italy, Einstein’s father and uncle were running an innovative technological company making and installing generators, bringing electricity to businesses, public buildings, streets, and even households for the very first time. Today, of course, we rue the fact that most large generators have been coal-fired—in the absence of a natural force such as falling water or wind, you needed heat to produce steam, in order to turn the huge coils that generate the current. Still, few of us would want to do without electricity, so we owe thanks not just to the modern scientists and engineers designing sustainable, climate-safe ways of producing it, but also to the nineteenth-century experimentalists and mathematicians who were trying to figure out just how it all works.

当麦克斯韦首次踏上电磁之旅时，理清其中的数学问题变得相当棘手。这与在科学中使用类比的优势和陷阱有关——在本例中，类比是指重力、流体动力学和电力之间的类比。我已经提到过重力和静电之间的数学相似性，但流体动力学——研究流体在力的作用下如何表现的学科——之所以重要，是因为在早期，人们倾向于将电视为流体。这就是为什么他们用“电流”这个词来描述“带电流体”的流动，不管它是什么：电子直到十九世纪末才被发现。但麦克斯韦意识到，很少有人意识到这一点，类比可能会让你误入歧途——事实上，电根本不是一种流体。不过，他知道，至少在最初，这种类比为理解电磁学这一新现象的数学提供了一种及时而有力的方法。挑战在于使用数学而不将其与物理现实混淆。

When Maxwell first set out on his electromagnetic journey, sorting out the maths of it had become rather tricky. It had to do with the advantages and pitfalls of using analogies in science—in this case, between gravity, hydrodynamics, and electricity. I’ve mentioned already the mathematical parallels between gravity and static electricity, but hydrodynamics—the study of how fluids behave as a result of forces—was important because in those early days people tended to visualise electricity as a fluid. That’s why they used the word “current” to describe the flow of “charged fluid,” whatever it was: the electron wouldn’t be discovered until the very end of the nineteenth century. But Maxwell realised, as very few had done, that analogies could lead you astray—indeed, electricity turned out not to be a fluid at all. Still, he knew that initially, at least, this analogy had offered a timely and powerful way into the mathematics of the new phenomenon of electromagnetism. The challenge was to use the maths without confusing it with the physical reality.

例如，另一个与流体有明显联系的术语是“通量”。牛津英语词典将其定义为“流动或流出的过程”。但在 18 世纪和 19 世纪初的流体研究中，“通量”获得了更具体的含义，即在给定时间内流过给定表面（例如管道的横截面积）的流体量。你可以把它想象成每秒计算离开管道的流体分子数，你可以用质量或体积来测量这个数字。在图 6.1中，我用体积显示了通量。

For instance, another term that has evident links with fluids is “flux.” The Oxford English Dictionary defines it as “a process of flowing or flowing out.” But in the study of fluids in the eighteenth and early nineteenth centuries, “flux” acquired a more specific meaning, as the amount of fluid flowing through a given surface—such as the cross-sectional area of a pipe—in a given time. You can think of it as counting the number of molecules of fluid leaving the pipe each second—and you can measure this number using mass or volume. In figure 6.1, I’ve shown the flux using volume.

一旦你做出了这样的定量定义，你就可以对其进行数学定义。对于流过圆柱形管道的流体，通量就是管道横截面积和流体速度的标量积（如图6.1 的标题所示）。对于流过任何形状表面的通量，而不仅仅是具有简洁面积公式的圆，其定义涉及表面积分。这是因为你实际上是在将流过表面上每个小面积元素的液体量相加（积分）。

Once you make a quantitative definition like this, you can define it mathematically. For fluid flowing through a cylindrical pipe, the flux turns out to be the scalar product of the cross-sectional area of the pipe and the velocity of the fluid (as you can see in the caption to fig. 6.1). In the case of flux through a surface of any shape, not just a circle with its neat area formula, the definition involves a surface integral. That’s because you’re effectively adding up (integrating) the amount of liquid flowing across each little element of area on the surface.

所有这些都表明，通量是任何矢量的属性，而不仅仅是流体的速度。果然，正如库仑静电定律和牛顿引力定律具有相同的数学形式，正如拉格朗日势能和拉普拉斯-泊松方程可以应用于电和重力一样，事实证明，通量积分可以应用于流体以外的其他事物——包括重力、电和磁。高斯是发展这种数学的先驱之一，因此将通过封闭表面的引力和电通量分别与封闭的质量和电荷量联系起来的定律被称为“高斯定律”。事实上，你可以从高斯定律推导出牛顿定律和库仑定律——而且磁通量也有高斯定律。不过，这里的重点是，所有这些早期的通量应用都涉及积分。但物理上，真正有趣的事情发生在这些电和磁通量发生变化，因为此时电磁效应开始发挥作用。因此，这些变化的磁通量是电磁学实际应用的关键，如图 6.2中电动机和发电机的示意图所示。

All this suggests that flux is a property of any vector, not just the velocity of a fluid. And sure enough, just as Coulomb’s law of static electricity and Newton’s law of gravity have the same mathematical form, and just as Lagrange’s potential and the Laplace-Poisson equations can be applied to both electricity and gravity, so it turned out that flux integrals could be applied to other things than fluids—including gravity, electricity, and magnetism. Gauss was one of the pioneers in developing this maths, so the laws relating the gravitational and electrical flux through a closed surface to the enclosed amount of mass and charge, respectively, are called “Gauss’s laws.” In fact, you can deduce Newton’s and Coulomb’s laws from Gauss’s laws—and there’s a Gauss’s law for magnetic flux, too. The main point here, though, is that all these early applications of flux involved integrals. But physically, the really interesting things happen when these electric and magnetic fluxes change, for that’s when electromagnetic effects come into play. So, these changing fluxes hold the key to practical applications of electromagnetism, as in the schematic setup for electric motors and generators in figure 6.2.

许多研究人员都在试图弄清这一现象究竟是如何发生的，以及如何用数学表达出来，其中包括法国的安培和奥古斯丁·柯西、英国的格林和汤姆森以及德国物理学家威廉·韦伯。麦克斯韦阅读了他们的所有著作，并钦佩他们的各种发现和假设。然而，这些杰出人物中没有一个人能够找到完整的电磁理论。

Exactly how this happened, and how it could be expressed mathematically, is what a host of researchers were trying to find out, including Ampère and Augustin Cauchy in France, Green and Thomson in Britain, and the German physicist Wilhelm Weber. Maxwell read up on them all and admired their various discoveries and hypotheses. Yet none of these luminaries had managed to find a complete theory of electromagnetism.

现在我们开始看到麦克斯韦的哲学训练在哪里脱颖而出——更不用说他的天才了。虽然不同物理现象之间的类比当然非常有用，但它们带有隐藏的假设——麦克斯韦意识到，他的前辈选择的数学工具，包括所有这些积分，都体现了最大的假设。因为大多数研究人员都假设引力、电力、磁力和电磁力都是超距作用的——瞬间直接从一个物体跳跃到另一个物体，一个带电粒子或一个磁铁跳跃到另一个磁铁，以及从移动的磁铁跳跃到导线或从载流导线跳跃到磁铁。麦克斯韦是唯一一个清楚地认识到，我在这里谈论的通量和其他积分被认为是体现超距作用的人，因为它们只关注构成积分极限或积分边界的点、线和面。例如，重力或电力F将物体从点r ₁移动到点r _{2 时}所做的功为${\text{∫}}_{r_1}^{r_2}F\times dr$；进行这样的积分时，只需要极限积分的极限——你不需要知道r ₁和r ₂之间的路径。15所以你可以看到，对于那些相信所有作用都发生在两点、从一个点到另一个^点瞬间相互作用的人来说，这种类型的积分是正确的工具。再举一个例子，你只需要在（现代版）牛顿证明中积分的极限，即两个球形物体（例如太阳和行星）之间的引力吸引力就好像它们所有的质量都集中在它们的中心一样。你可以明白为什么这表明了超距作用：就好像这两个中心点之间没有发生任何引力一样。麦克斯韦意识到，同样，他的前辈使用的所有电磁积分都没有考虑可能发生的任何过程，不仅仅是在边界，而且是在中间或周围空间。也许这就是为什么没有人能够破解电磁密码的原因。

Now we begin to see where Maxwell’s training in philosophy comes to the fore—not to mention his genius. While analogies between different physical phenomena can certainly be very useful, they come with hidden assumptions—and Maxwell realised that his forerunners’ choice of mathematical tools, with all those integrals, embodied the biggest assumption of all. For most researchers assumed that gravity, electricity, magnetism, and electromagnetism all acted at a distance—instantaneously leaping directly from one material body to another, one charged particle, or one magnet, to another, as well as from a moving magnet to a wire or from a current-carrying wire to a magnet. Maxwell was the only one who recognised clearly that the flux and other integrals I’ve been talking about here were assumed to embody action-at-a-distance, because they focussed only on the points, lines, and surfaces that formed the limits of integration or boundaries of the integral. For instance, the work done by a gravitational or electric force F in moving an object from point r₁ to point r₂ is given by ${\text{∫}}_{r_1}^{r_2}F\times dr$; when you carry out such an integral, you need only the limits of integration—you don’t need to know anything about the path between r₁ and r₂.¹⁵ So you can see that this type of integral was the right tool for those who believed all the action happened at the two points, interacting instantaneously from one point to the other. To take another example, you only need the limits of integration in (the modern version of ) Newton’s proof that the gravitational attraction between two spherical objects, such as the Sun and a planet, acts as if all their mass is concentrated at their centres. You can see why this suggested action at a distance: it’s as if nothing gravitational happened in between these two central points. Maxwell realised that, similarly, all those electromagnetic integrals his predecessors were using took no account of any processes that might be going on not just at the boundaries, but also within the intervening or surrounding space. Perhaps that was why no one had managed to crack the electromagnetic code.

但这一切即将改变。经过一年的认真研究，麦克斯韦准备抛开主流数学家的著作，接受一个完全外行人提出的想法。

But all this was about to change. After his year of careful study, Maxwell was ready to set aside the works of the mainstream mathematicians, and to take up an idea suggested by a rank outsider.

因此，尽管法拉第不太懂数学，但他对电磁的物理性质的了解肯定比大多数人都要多——这促使他开发出一种独特的方法来研究电荷和磁铁之间空间中发生了什么。他第一次产生这个想法是因为他注意到了一件可能也会让你着迷的事情，如果你上过学校的科学课的话：铁屑整齐地排列成环状，围绕着条形磁铁的两极。这是因为每个小铁屑都会将自己的小极点朝向条形磁铁的极点，如图 6.3a所示——正如麦克斯韦后来解释的那样，铁屑线之所以形成，是因为每次添加新的铁屑时，它也必须与相邻的铁屑首尾相连，因为吸引磁力集中在每个小磁铁末端附近的极点上^。16

So, while Faraday didn’t know much maths, he certainly knew more than most about the physical nature of electromagnetism—and this led him to develop a unique approach to the question of what was going on in the space between electric charges and magnets. He first got the idea when he noticed something that may well have fascinated you, too, if you took a school science class: the neat way that iron filings line up in loops around the poles of a bar magnet. That’s because each little filing orients its own tiny poles to those of the bar magnet, as in figure 6.3a—and as Maxwell later explained, lines of filings are formed because each time a new filing is added, it must also align itself end to end with its neighbor, for the attracting magnetic force is concentrated at the poles near the ends of each little magnet.¹⁶

传统的“超距作用”理论认为，如果只有一块铁屑（如图 6.3b所示，一小块“测试”铁屑），那么它和磁铁之间的力将瞬间相互作用。换句话说，磁力只会在磁铁和铁屑之间起作用，而且会瞬间起作用。但法拉第推断，即使你把所有其他铁屑都拿走，磁铁的力仍然会像以前一样作用在所有相同的位置，即使你再也看不到它的影响。特别是，一定有力一直作用在磁铁和“测试”铁屑之间的空间，就像所有铁屑都存在时一样。为了描述这个想法，法拉第提出了“力线”的概念，与铁屑线直接类比。就好像铁屑使磁铁发出的不可见的力显现出来，就像光使水印可见一样。

The traditional, “action-at-a-distance” way of looking at this was that if you had just one iron filing present—a single little “test” filing, as in figure 6.3b—then the force between it and the magnet would interact instantaneously. In other words, the magnetic force would act only between the magnet and the filing, and it would do so instantaneously. But Faraday reasoned that even when you’d taken all the other iron filings away, the force from the magnet was still acting in all the same places as it was before— even if you could no longer see its effects. In particular, there must be force acting all the way through the space between the magnet and the “test” filing—just as there’d been when all the filings were present. To describe this idea, Faraday came up with the concept of “lines of force,” in direct analogy with the lines of iron filings. It was as if the filings made manifest the invisible force emanating from the magnet, just as light makes a watermark visible.

铁屑围绕磁铁排列的方式让人想起一片麦田。当微风吹过麦田时，每根小麦秆都会因风力而弯曲。同样，法拉第提出，电力和磁力都是通过带电粒子和磁铁产生的“场”传递的——他宣称，它们不会在远处起作用。这是一个简单而绝妙的想法。问题是，他没有知道如何将其转化为可预测的数学理论，以便通过进一步的实验进行检验。因此，尽管他凭借其无可否认的熟练的实验研究立即获得了主流赞誉，但他的电场和磁场概念却几乎被忽视了。

The way iron filings line up around a magnet brings to mind something like a field of wheat. When a breeze blows across the field, each little stalk bends in response to the force of the wind. Similarly, Faraday proposed that both electric and magnetic forces were conveyed through “fields” emanating from charged particles and magnets—they did not, he declared, act at a distance. It was a simple, utterly brilliant idea. Trouble was, he didn’t know how to turn it into a predictive, mathematical theory that could be tested against further experiments. So, while he’d gained immediate mainstream acclaim for his undeniably skillful experimental research, his concept of electric and magnetic fields was all but ignored.

在早期对热和流体流动的研究中，傅立叶、牛顿和欧拉等人实际上已经直观地想到了场的概念，尽管这是一种隐含的方式。但他们确实认为流体的每个粒子都有速度（和温度），这类似于法拉第的想法，即在带电体或磁铁周围的空间中，每个点的假设单元测试电荷和磁屑都会受到电力或磁力的作用。实际上，法拉第对力线的理解比对场的理解更清晰，汤姆森是物理学中第一个定义这一概念的人：只要空间中的每一点都有确定的磁力，他就会在 1851 年说，你就有一个“磁力场”，或者更简单地说，“一个磁场”。当然，“电场”也是如此^。17

In early studies of heat and fluid flow, the likes of Fourier, Newton, and Euler had, in fact, intuited the idea of a field, albeit it an implicit way. But they did have the idea that each particle of fluid has a velocity (and a temperature), which is similar to Faraday’s idea that there are electric or magnetic forces acting on hypothetical unit test charges and filings placed at each point in the space around an electrically charged body or a magnet. Actually, Faraday had a clearer idea about lines of force than he did about the field, and it was Thomson who first defined the concept in physics: whenever there is a definite magnetic force at every point in a space, he’d said in 1851, you have a “field of magnetic force,” or more simply, “a magnetic field.” And similarly, of course, for an “electric field.”¹⁷

麦克斯韦将他的论文（第一篇关于电的论文）命名为“论法拉第的力线”，并在 1855-56 年冬天将其分两部分提交给剑桥哲学学会。（1855 年，他终于获得了剑桥的奖学金！）论文发表后，他寄了一份给法拉第。64 岁的法拉第已经厌倦了主流对他的场理论的拒绝，因此你可以想象当他发现 24 岁的麦克斯韦终于将其付诸实践时，他有多么高兴。法拉第热情地回复了麦克斯韦，两人成为了朋友。

Maxwell called his paper—his first on electricity—“On Faraday’s Lines of Force,” and he presented it to the Cambridge Philosophical Society in two parts over the winter of 1855–56. (In 1855 he’d finally secured a fellowship at Cambridge!) When it was later published, he sent a copy to Faraday. At the age of sixty-four, Faraday had grown weary of the mainstream’s rejection of his field idea, so you can imagine his joy on finding that twenty-four-year-old Maxwell had brought it to life at last. Faraday replied warmly, and the two men became friends.

当时，麦克斯韦还在照顾他深爱的父亲，他的父亲于 1856 年 4 月去世。他将自己的悲痛写进一首诗，然后尽自己所能，继续担任格伦莱尔的新领主，履行所有相关职责。（当然，这也是他的乐趣所在：麦克斯韦一直与庄园的佃户相处融洽，他和父亲还为那些想要提高阅读能力的人设立了一个项目。）几周后，他终于获得了阿伯丁马歇尔学院的教授职位，击败了亚瑟·凯莱（凯莱在当了多年律师后正试图重返学术界）和泰特。18^两年后，他与学院校长的女儿凯瑟琳·杜瓦结婚。和汉密尔顿的妻子海伦一样，凯瑟琳也非常虔诚；麦克斯韦和汉密尔顿一样，对精神有着更广泛的看法，尽管他们都是虔诚的基督徒。

Maxwell had also been nursing his adored father, who died in April 1856. He poured his grief into a poem and then carried on as best he could as the new Laird of Glenlair, with all its attendant duties. (And its pleasures: Maxwell had always got on well with the estate’s tenants, and he and his father had set up a program for those who wanted to improve their reading.) A few weeks later, he finally landed a professorial appointment, at Marischal College in Aberdeen, beating Arthur Cayley to the job— Cayley was trying to get back into academia after his years as a lawyer— and Tait, too.¹⁸ Two years later he married the college principal’s daughter, Katherine Dewar. Like Hamilton’s wife, Helen, Katherine was strictly religious; Maxwell, like Hamilton, had a broader view of spirituality, although they were both committed Christians.

在接下来的十年里，麦克斯韦不断完善他对法拉第场的概念，并发表了许多其他论文，直到最后，他准备发表他的电磁场理论。它包括我提到的势能和通量的概念——但它包含的积分相对较少。相反，它有很多导数和“偏微分方程”——涉及关于x、y和z的导数的方程，比如我之前展示的拉普拉斯算子中势能的导数。

Many other papers followed in the next decade as Maxwell honed his conception of Faraday’s field until, at last, he was ready to publish his radical field theory of electromagnetism. It includes the ideas of potential and flux that I’ve mentioned—but it contains relatively few integrals. Instead, there are many derivatives and “partial differential equations”—equations that involve derivatives with respect to x, y, and z, such as the derivatives of the potential in the Laplacian I showed earlier.

当然，他并不是唯一一个使用微分方程的人——自牛顿以来，人们就一直在物理学中使用它们。只是他比任何人都更清楚何时使用每种微积分，以及为什么使用。他仔细解释了他对电磁学中导数的偏爱，并指出，虽然线积分和曲面积分只关注诸如两个相互作用粒子之间的距离和“这些物体中的电或电流”之类的东西——如果你假设电力和磁力在远处作用，那么这是可以的——但偏微分方程显示了物体周围整个空间的变化。因此，他认为，这样的方程是表达变化的电力和磁力和通量产生电磁效应的方式的自然工具。场，无论它是由什么构成的，都通过空间介导这些变化——就像麦田介导像波浪一样穿过它的微风一样^。19

He wasn’t the only one to use differential equations, of course—people had been using them in physics ever since Newton. It’s just that he was clearer than anyone else about when to use each kind of calculus, and why. He explained his preference for derivatives in electromagnetism carefully, noting that while line and surface integrals focus only on such things as the distance between the two interacting particles and the “electrifications or currents in these bodies”—which was fine if you assumed that electric and magnetic forces acted at a distance—partial differential equations show how things change through the whole space around the bodies. So, he suggested, such equations are the natural tools for expressing the way changing electric and magnetic forces and fluxes produce electromagnetic effects. The field, whatever it was made of, mediated these changes through space—just as the field of wheat mediates a breeze rippling through it like a wave.¹⁹

• • •

• • •

麦克斯韦是第一个将法拉第发现的磁感应（即相对运动的磁铁感应出电流）转化为符号数学形式的人。将磁铁穿过线圈或将线圈穿过磁场会改变磁通量（或者法拉第认为是从磁铁穿过线圈的“力线”数量，如图6.2所示）；麦克斯韦必须找到正确的方程来将这种变化的磁通量与感应电流量联系起来。这今天被称为“法拉第定律”。

Maxwell was the first to put Faraday’s discovery of magnetic induction— the inducing of electric current by a relatively moving magnet—into symbolic mathematical form. Moving the magnet through a coil of wire, or turning the coil through a magnetic field, changes the magnetic flux—or what Faraday thought of as the number of “lines of force” from the magnet through a loop of wire, as you can see in figure 6.2; Maxwell had to find the right equations for relating this changing magnetic flux to the amount of current induced. This is known today as “Faraday’s law.”

然后是“安培定律”，这是奥斯特发现的数学版本，用变化的电流产生的磁力来表示。麦克斯韦通过更仔细地定义电流，然后展示它与变化的电通量的关系，扩展了安培在这方面的工作。20^我之前谈到通量是表面积分，安培也使用过表面（和线）积分，因为他一直从事超距作用传统。因此，麦克斯韦不仅需要完成安培的结果，还需要将这些积分重写为导数，以表达场的概念。

Then there was “Ampère’s law,” the mathematical version of Øersted’s discovery, in terms of the magnetic force produced by a changing electric current. Maxwell extended Ampère’s work on this by defining current more carefully, and then showing how this relates to changing electric flux.²⁰ I spoke before about flux as a surface integral, and Ampère had used surface (and line) integrals, too, for he had worked in the action-at-a-distance tradition. So, Maxwell needed not only to complete Ampère’s results but also to rewrite these integrals as derivatives, in order to express the field concept.

还有许多内容需要描述和定义，包括静电和磁力的高斯-库仑定律，麦克斯韦现在必须用场的语言重写它们。即使电荷或磁铁静止不动，平方反比定律也会显示它们各自发出的力如何随距离而变化（“静态”表示力不会随时间而变化）——因此，导数也是这些定律的自然语言选择。在条形磁铁周围的磁场中，你可以看到铁屑排列的变化，铁屑聚集在力最强的两极附近，但随着距离的增加而减弱，铁屑向外扩散——如图6.3a和6.3c所示。

There was much more to describe and define—including the Gauss-Coulomb laws for static electricity and magnetism, which Maxwell now had to rewrite in field language. For even when you have a charge or a magnet just sitting still, the inverse square law shows how the force emanating from each of them changes with distance (“static” means the force doesn’t change in time)—so derivatives are a natural choice of language for these laws, too. In the magnetic field around a bar magnet, you can see these changes in the alignment of iron filings, the filings bunching up near the poles where the force is strongest but spreading out as it weakens with distance—as in figures 6.3a and 6.3c.

麦克斯韦从传统的超距作用积分转变为偏微分场方程的核心在于“积分学基本定理”，你可能还记得学校里学过这个定理，不过我会在下一个尾注中详细说明。你也会从中了解到麦克斯韦对斯托克斯定理的巧妙运用，该定理曾出现在史密斯奖考试中——你会看到他是如何运用现在的后矢量的，被称为“散度定理”，由高斯、格林和俄罗斯数学家米哈伊尔·奥斯特拉格拉德斯基首创^。21

The nub of how Maxwell changed from traditional action-at-a-distance integrals to partial differential field equations lies in the “fundamental theorem of integral calculus,” which you may remember from school, although I’ll spell it out in the next endnote. You’ll also get a hint there of the marvelous use Maxwell made of Stokes’s theorem, which had featured in the Smith’s Prize exam—and you’ll see how he used what is now, post-vectors, known as the “divergence theorem,” pioneered by Gauss, Green, and the Russian mathematician Mikhail Ostragradsky.²¹

我们将在下一章中看到麦克斯韦方程组，但这里的重点是，首先，在他关于电磁场的里程碑式论文中——他在 1864 年底提交给皇家学会，并于 1865 年 1 月在其《哲学学报》上发表——他首次正式统一了电和磁。

We’ll see Maxwell’s equations in the next chapter, but the point here is that first, in his landmark paper on the electromagnetic field—which he presented to the Royal Society at the end of 1864 and published in its Philosophical Transactions in January 1865—he formally united electricity and magnetism for the first time.

其次，与高斯、斯托克斯等人一样，麦克斯韦只使用了斯托克斯定理和散度定理的分量形式（他没有以当今教科书中的紧凑全矢量形式来陈述它们）。因此，在他 1865 年的论文中，他发现了他为描述电磁场而定义的各种量的分量之间的关系。换句话说，他还没有以完整的矢量形式阐述他的理论。不过，他确实谈到了既有方向又有大小的量，所以他创建了一个矢量电磁理论。在泰特的帮助下，他只用了几年时间就得到了汉密尔顿全矢量微积分。

Second, like Gauss, Stokes, et al., Maxwell used only the component form of Stokes’s theorem and the divergence theorem (he didn’t state them in the compact whole-vector form you see in textbooks today). So, in his 1865 paper, he found relationships between the components of the various quantities he’d defined to describe the electromagnetic field. In other words, he had not yet formulated his theory in full vector form. He did talk about quantities having both direction and magnitude, though, so he had created a vectorial electromagnetic theory. With Tait’s help, he would get to Hamiltonian whole-vector calculus in just a few years’ time.

1865 年是美国历史上以及物理学和数学史上具有重要意义的一年，因为血腥的内战终于结束了。帮助转化麦克斯韦方程的人之一最早将现代（后汉密尔顿）向量符号引入到现代向量符号的人是约西亚·威拉德·吉布斯，他的父亲长期以来一直致力于废除奴隶制。战争期间，年轻的吉布斯在耶鲁大学攻读工程学博士学位——这是美国第一个工程学博士学位。又过了十年，他才发现麦克斯韦，又过了几乎十年，他才在向量史上留下自己的印记。所以，我在这个故事中试图展示的部分内容是，思想的发展和找到最佳形式需要多长时间。我认为，了解这段漫长的旅程可以帮助那些难以掌握一个想法的学生，因为他们会发现早期的数学家也曾经历过挣扎。但它也可以帮助展示我们现在可能认为理所当然的具体思想的力量——而在认为它们理所当然的过程中，我们可能会错过其中的一些力量。

The year 1865 is a momentous one in American history as well as in physics and maths history, for the bloody Civil War finally ended. It is fitting that one of the men who would help transform Maxwell’s equations into modern (post-Hamilton) vector notation is Josiah Willard Gibbs, whose father had long agitated for the abolition of slavery. During the war, young Gibbs was completing his engineering doctorate at Yale—the very first American engineering doctorate. It would be another decade before he discovered Maxwell, and almost another still before he made his mark in the history of vectors. So, part of what I’ve been trying to show throughout this story is just how long it takes for ideas to develop and find their best form. I think that knowing something of this long journey can help students who are struggling to grasp an idea, by showing that earlier mathematicians had struggled, too. But it can also help show the power of specific ideas that we might now take for granted—and in taking them for granted, we might miss some of that power.

例如，麦克斯韦选择用微分方程来表示电磁行为，但这些方程在数学上等同于早期电磁先驱们如果成功的话可能开发的积分方程——事实上，麦克斯韦方程的积分形式今天经常使用。（这种等价性正是上一个尾注中的“基本定理”的用武之地。）区别在于解释，尽管今天，无论你使用哪种形式的微积分，场的概念在物理学中都是至关重要的——正如法拉第、汤姆森和麦克斯韦所表明的那样，“场”只是特定量在空间中的值分布。例如，你周围空气温度的记录是一个“标量场”——一组表示空间中每个点温度大小的数字——而作用于整个场中小单位电荷或铁屑的电力和磁力是“矢量场”，因为在每个点，这些力都有大小和方向。 （换句话说，矢量和矢量场之间的区别在于，矢量在给定点处定义，而矢量场为相关空间中的每个点分配一个矢量。）但麦克斯韦选择了微积分，这不仅使他能够从数学上定义电磁场，而且还做出了令人惊讶且影响深远的发现——这项发现彻底改变了我们彼此以及与宇宙本身的交流方式。

For instance, Maxwell chose to represent the behavior of electromagnetism using differential equations, but these are mathematically equivalent to the integral equations that the early electromagnetic pioneers might have developed had they succeeded in their quest—in fact, the integral form of Maxwell’s equations is often used today. (This equivalence is where the “fundamental theorem” in the previous endnote comes into it.) The difference is in the interpretation, although today, the idea of a field is paramount in physics, no matter which form of calculus you use—for as Faraday, Thomson, and Maxwell showed, a “field” is just the distribution of values of a particular quantity through space. For example, a record of the temperature of the air around you right now is a “scalar field”—a set of numbers representing the magnitude of the temperature at each point in space—while the electric and magnetic forces acting on little unit test charges or filings placed throughout the field are “vector fields,” because at each point these forces have both magnitude and direction. (In other words, the difference between a vector and a vector field is that a vector is defined at a given point, and a vector field assigns a vector to each point in the relevant space.) But it was Maxwell’s choice of differential calculus that enabled him not only to define electromagnetic fields mathematically, but also to make a surprising and far-reaching discovery—one that has revolutionised the way we communicate with each other, and with the universe itself.

更重要的是，麦克斯韦发现他理论中的电磁波也几乎与光速相同。（我说“几乎”是因为当时很难进行精确测量。）这是一个不容忽视的巧合——它证明了麦克斯韦对导数的直觉既深刻又简单。最初的横向“波动方程”是一个微分方程，一个世纪前就被推导出来描述拨弦的振动方式——但这个引人注目的新数学平行表明，正如麦克斯韦谦虚地说的那样，“光本身（包括辐射热，以及其他辐射，如果有的话）是一种电磁扰动，以波的形式通过电磁场传播，遵循电磁定律。”他一举将电和磁统一起来，也把光统一起来了。他甚至指出了“其他辐射”的可能存在，这一诱人的预测让海因里希·赫兹等人走上了发现无线电波的道路。

What’s more, Maxwell found his theoretical electromagnetic waves had just about the same speed as light, too. (I say “just about” because of the difficulty in making accurate measurements at the time.) It was a coincidence too delicious to ignore—and it proved that Maxwell’s hunch about derivatives was as profound as it was simple. The original transverse “wave equation,” a differential equation, had been derived a century earlier to describe the way a plucked string vibrates—but this remarkable new mathematical parallel suggested, as Maxwell put it modestly, “that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.” In one fell swoop he had united not just electricity and magnetism but light, too. He’d even pointed to the possible existence of “other radiations,” a tantalising prediction that set the likes of Heinrich Hertz on the path to discovering radio waves.

正如爱因斯坦后来写道，想象一下麦克斯韦在电磁学和光之间建立联系时的感受！“世界上很少有人能有这样的经历，”他补充道。麦克斯韦本人在 1865 年 1 月写给他表弟的一封信中，以他一贯低调的方式私下表达了他的兴奋之情：他几乎是顺便补充说：“我还有一篇论文，里面有关于光的电磁理论，除非我确信相反，否则我认为它是伟大的武器。”他不喜欢吹嘘，所以你可以感觉到当他做出这个不可思议的推论时，他一定是多么激动^。22

As Einstein later wrote, imagine Maxwell’s feelings when he made the connection between electromagnetism and light! “To few men in the world has such an experience been vouchsafed,” he added. Maxwell himself had privately expressed his excitement in his typically understated way, in a letter to his cousin in January 1865: almost as an aside he’d added, “I have also a paper afloat, with an electromagnetic theory of light, which, till I am convinced to the contrary, I hold to be great guns.” He was not given to boasting, so you can sense how thrilling it must have been when he made his uncanny deduction.²²

麦克斯韦之所以能发现光的电磁性质，是因为他巧妙地选择了数学语言。23起初，人们并没有热烈地接受它。主流面临的问题是，法拉第的场的概念已经表达出来^了只有细致的数据，没有方程，现在麦克斯韦的理论完全用方程来表达——没有物理模型来理解场到底是什么。这种反应也曾困扰着牛顿的引力理论。在早期的多篇论文中，麦克斯韦确实提出了几种力学模型来解释场是什么以及场是如何传递电磁效应的，但他知道没有办法证明其中任何一个的存在。他相信，场的后果的数学知识应该足够了——就像牛顿知道，尽管他没有解释引力本身，但只要他的方程准确地描述引力的效应就足够了。

Maxwell’s momentous discovery of the electromagnetic nature of light was only possible because of his clever choice of mathematical language.²³ Not that it met with rapturous acceptance at first. The problem for the mainstream was that Faraday’s conception of the field had been expressed with meticulous data but no equations, and now Maxwell’s theory was expressed entirely in terms of equations—with no physical model for conceiving what the field actually was. It was the very same response that had dogged Newton’s theory of gravity. In various earlier papers, Maxwell had indeed suggested several mechanical models that might explain what the field was and how it transmitted electromagnetic effects, but he knew there was no way of proving the existence of any of them. The mathematics of the field’s consequences, he believed, should be enough—just as Newton had known it was enough that his equations accurately described the effects of gravity, even though he hadn’t explained gravity itself.

尽管如此，麦克斯韦仍在思考如何更清楚地表达他的场。在 1865 年的论文开头，他说找到电磁理论的第一步是“确定物体之间作用力的强度和方向”——就像牛顿最初开始定义力时所做的那样。这是一种矢量方法，但还不是完整的矢量理论，因为麦克斯韦还没有研究四元数。但这种情况即将改变——在他的老朋友泰特的帮助下。

Still, Maxwell kept on thinking about how he could express his fields more clearly. At the beginning of his 1865 paper, he’d said that the first step in finding a theory of electromagnetism had been to “ascertain the strength and direction of the forces acting between the bodies”—just as Newton had done when setting out to define force in the first place. It was a vectorial approach, but not yet a full vector theory, for Maxwell hadn’t yet studied quaternions. But that was about to change—with help from his old friend Tait.

如果你学过矢量微积分，你就会知道这是

If you’ve studied vector calculus, you’ll recognise this as

$\nabla =\frac{\partial }{\partial 十}我+\frac{\partial }{\partial 是}杰+\frac{\partial }{\partial 是}钾$

$\nabla =\frac{\partial }{\partial x}i+\frac{\partial }{\partial y}j+\frac{\partial }{\partial z}k$

用今天的符号表示。（正如我在序言中提到的，还有其他现代的方式来表示向量，比如在字母顶部加帽子或横线，或者在下面加波浪线，但粗体既醒目又容易输入，所以在现代教科书和论文中很常见。）无论哪种方式，你处理的都是笛卡尔坐标系，所以我们很快就会看到，无论你使用单位虚数i、j、k还是单位向量i、j、k作为向量运算符的基础，你基本上都会得到相同的结果。

in today’s notation. (As I mentioned in the prologue, there are other modern ways of denoting vectors, such as hats or bars on top of a letter or squiggles underneath, but boldface is both arresting and easy to type, so it is common in modern textbooks and papers.) Either way, though, you’re dealing with a Cartesian coordinate system, so we’ll see shortly that you essentially get the same results, no matter whether you use the unit imaginary numbers i, j, k or the unit vectors i, j, k as the basis of your vector operator.

泰特立即着手将这种新型算符应用到他的热和电研究中，并将汉密尔顿的符号 ⊲ 变成了 ∇。泰特还为这个符号取了一个名字“nabla”，这是他的助手威廉·罗伯逊·史密斯建议的，因为“nabla”是希腊语中古亚述竖琴的拉丁音译，其形状与 ∇ 类似，呈倒三角形。（后来，吉布斯将汉密尔顿的“nabla”替换为“del”——显然是因为 ∇ 也类似于希腊字母“delta”的倒置版本。这两个名字今天仍在使用。）至于现代表示 ∇ 的粗体字，当反传统的奥利弗·赫维赛德最终进入我们的故事时，它将载入史册。但我在这里介绍它，是因为我认为它可以让现代读者更容易地看到，正如泰特所做的那样，汉密尔顿的算符本身是矢量的。

Tait immediately set to work applying this new kind of operator in his work on heat and electricity, turning Hamilton’s symbol ⊲ into ∇ in the process. Tait also linked this symbol with a name, “nabla,” which his assistant William Robertson Smith suggested, because “nabla” is the Latin transliteration of the Greek word for an ancient Assyrian harp, which had an inverted triangle shape similar to ∇. (Later, Gibbs will replace Hamilton’s “nabla” with “del”—apparently because ∇ also resembles an upside-down version of the Greek letter “delta.” Both names are used today.) As for the boldface type in the modern representation of ∇, it will make its way into history when the iconoclastic Oliver Heaviside eventually enters our story. But I’m introducing it here because I think it makes it easier for modern readers to see, as Tait certainly did, that Hamilton’s operator itself is vectorial.

例如，这意味着它可以将普通（标量）函数（例如势）转换为矢量。前面我提到，如果力具有势V，则力的笛卡尔分量为$\frac{\partial 五}{\partial 十}，\frac{\partial 五}{\partial 是}，\frac{\partial 五}{\partial 是}$。这些只是您插入后获得的组件$\nabla =我\frac{\partial }{\partial 十}+杰\frac{\partial }{\partial 是}+钾\frac{\partial }{\partial 是}$ 或变成现代矢量版本，$\nabla =\frac{\partial }{\partial 十}\mathit{我}+\frac{\partial }{\partial 是}\mathit{杰}+\frac{\partial }{\partial 是}\mathit{钾}$。因此，我们不必通过列出力的组成部分来写力，

This means, for example, that it can turn an ordinary (scalar) function, such as the potential, into a vector. Earlier I mentioned that if a force has a potential V, the Cartesian components of the force are $\frac{\partial V}{\partial x},\frac{\partial v}{\partial y},\frac{\partial V}{\partial z}$. These are just the components you get if you insert $\nabla =i\frac{\partial }{\partial x}+j\frac{\partial }{\partial y}+k\frac{\partial }{\partial z}$ or into the modern vector version, $\nabla =\frac{\partial }{\partial x}\mathit{i}+\frac{\partial }{\partial y}\mathit{j}+\frac{\partial }{\partial z}\mathit{k}$. So instead of writing the force by listing its components,

$F=\left(\frac{\partial 五}{\partial 十}，\frac{\partial 五}{\partial 是}，\frac{\partial 五}{\partial 是}\right)，$

$F=\left(\frac{\partial V}{\partial x},\frac{\partial V}{\partial y},\frac{\partial V}{\partial z}\right),$

你可以更简单、更透明地写成

you can write it much more simply and transparently as

F = ∇ V，

F = ∇V,

或F = ∇ V，正如 Tait 和后来的麦克斯韦所写。

or F = ∇V as Tait, and later Maxwell, wrote it.

它更简单，因为它是一种更紧凑的表示F的方式，在编写或编程时节省时间和空间。它更透明，因为它将力表示为一个整体矢量，而不是单独列出其分量。这使得更容易看出问题是关于力和势能的物理概念，而不仅仅是数字（分量）值的列表。这是一个微妙的点，我们故事中的一些玩家需要很长时间才能理解它。

It’s simpler because it’s a more compact way of representing F, saving time and space when you are writing it out or programming it. And it’s more transparent because it expresses the force as a whole vector, rather than listing its components separately. This makes it easier to see that the problem is about force and potential as physical concepts, rather than just a list of numerical (component) values. This is a subtle point, and it will take some of the players in our story a long time to understand it.

抛开物理学不谈，由于 ∇ 是矢量，你也可以将其与自身形成标量积，将拉普拉斯算子写为 ∇ ∙ ∇，或者哈密顿所写的 ∇ ²，我们今天也这样做。因此，例如，拉普拉斯方程变成

Physics aside, since ∇ is vectorial, you can also form its scalar product with itself, writing the Laplacian as ∇ ∙ ∇, or ∇² as Hamilton wrote it, and as we also do today. So, for example, Laplace’s equation becomes

∇2V = ⁰，

∇²V = 0,

这比$\frac{{\partial }^2五}{\partial {十}^2}+\frac{{\partial }^2五}{\partial {是}^2}+\frac{{\partial }^2五}{\partial {是}^2}=0$。

which is neater than $\frac{{\partial }^{2}V}{\partial x^2}+\frac{{\partial }^{2}V}{\partial y^2}+\frac{{\partial }^{2}V}{\partial z^2}=0$.

你也可以取 nabla 和另一个向量的标量积——也可以取向量或叉积。下一章中我们将看到更多这方面的内容。这将使 Tait 能够使用这些 nabla 运算以全向量微积分的形式写出斯托克斯定理和散度定理——他将是第一个这样做的人。不过，他首先全身心地投入到汉密尔顿的书中。他对四元数及其向量微积分算子在物理学中的作用感到非常兴奋，因此他问汉密尔顿是否介意与他通信。汉密尔顿当然很高兴找到了一个数学上的志趣相投的人。在 1858 年他们开始通信时，他告诉 Tait那是一个奇妙的日子，四元数的秘密在布鲁姆桥上一闪而过，“我的孩子们从此就把这座桥叫做四元数桥。” ²⁴

You can also take the scalar product of nabla and another vector—and you can take the vector or cross product, too. We’ll see more of this in the next chapter. It will enable Tait to use these nabla operations to write Stokes’s theorem and the divergence theorem in terms of whole-vector calculus—and he’ll be the first to do this. First, though, he threw himself into Hamilton’s book. He was so excited about the role that quaternions and their vector calculus operators could play in physics that he asked if Hamilton would mind corresponding with him. Hamilton, of course, was delighted to find a mathematical kindred soul. At the beginning of their correspondence in 1858, he told Tait of that marvelous day when the secret of quaternions revealed itself in a flash at Broome Bridge, “which,” he added, “my boys have since called the Quaternion Bridge.”²⁴

到了 1859 年，汉密尔顿已经可以自在地与泰特分享一些私人感受，就像他与德·摩根一样——包括讲述他的初恋凯瑟琳·迪斯尼嫁给别人时他所感到的绝望。（多年后，她告诉汉密尔顿，她只爱他一个人，她的父母强迫她接受一段更“合适”但最终不幸福的婚姻。）但正如他告诉泰特的那样，“那些日子已经过去了：幸福吗？是的，就变得更理智、更少感性而言。” ²⁵看来他读过简·奥斯汀的《理智与情感》！无论如何，54 岁的汉密尔顿很满足。

By 1859 Hamilton felt comfortable enough to share quite personal feelings with Tait, as he did with De Morgan—including recounting the despair he’d felt when his first love Catherine Disney had married someone else. (It was years later that she told Hamilton she had loved only him, and that her parents had pressured her into a more “suitable,” ultimately unhappy marriage.) But as he told Tait, “Those days are over: happily? Yes, so far as getting a little more sense, and less sensibility, is concerned.”²⁵ It seems he’d read Jane Austen’s Sense and Sensibility! At any rate, at the age of fifty-four, Hamilton was content.

28 岁的泰特似乎也很满足。两年前，他与玛格丽特·波特结婚，也陶醉于工作。除了在贝尔法斯特教书之外，他还在 1859 年发表了第一篇关于四元数的论文（论波）。然后，在 1860 年，他晋升为爱丁堡大学的自然哲学教授，当时他和麦克斯韦都就读于这所大学。麦克斯韦也申请过这个职位，但不管对错，人们都认为泰特更胜一筹。JM 巴里在成为《彼得潘》的作者之前是泰特的学生之一，他声称泰特是有史以来最出色的演示者。他回忆说，上课时，泰特“闪烁的小眼睛里闪烁着迷人的光芒……我曾看到过一个人在泰特的目光下惊慌失措。”然而，巴里继续说，那双眼睛也可以“像男孩一样快乐”——尤其是当“他把一管水倒在一群坚持挤在实验太近的学生身上时”。但麦克斯韦也有他的学生粉丝——例如，未来的天文学家大卫·吉尔描述了麦克斯韦在讲座结束后会和感兴趣的学生呆上几个小时。至于泰特和麦克斯韦本人，从他们之间的信件来看，这种职业竞争并没有影响他们的友谊。毕竟，他们从学生时代起就是朋友和对手^。26

Twenty-eight-year-old Tait seemed content, too. He had married Margaret Porter two years earlier, and he was also reveling in his work. As well as his teaching in Belfast, he published his first quaternion paper (on waves) in 1859. Then, in 1860, he won a promotion, to professor of natural philosophy at his and Maxwell’s old university, Edinburgh. Maxwell had applied for the post, too, but rightly or wrongly Tait was seen as the better lecturer. J. M. Barrie was one of Tait’s students, before he became famous as the author of Peter Pan, and he claimed that Tait was the most superb demonstrator ever. He recalled that during classes Tait’s “small twinkling eyes had a fascinating gleam in them…. I have seen a man fall back in alarm under Tait’s eyes.” Yet, Barrie continued, those eyes could also “be merry as a boy’s”—especially when “he turned a tube of water on a crowd of students who would insist on crowding too near an experiment.” But Maxwell had his student fans, too—for instance, the future astronomer David Gill described how Maxwell would stay behind with interested students for hours after the lecture. As for Tait and Maxwell themselves, such professional rivalry had no effect on their friendship, judging by their letters to each other. After all, they’d been friends and rivals since school.²⁶

我们将在下一章中看到他们之间古怪的通信，因为 Tait 即将因其在四元数方面的工作而出名，而 Maxwell 即将向他提出大量问题。他将在他的《电磁论》中使用答案——这本非凡的书将为现代矢量微积分之路的最后一步奠定基础。

We’ll see something of their quirky correspondence in the next chapter, for Tait is about to become famous for his work on quaternions, and Maxwell is about to pepper him with questions. He will use the answers in his Treatise on Electricity and Magnetism—and this extraordinary book will set the scene for the final step on the road to modern vector calculus.

泰特着迷于四元数（更确切地说，四元数的矢量部分）如何使他能够更紧凑地写出物理方程，尤其是当他应用哈密顿的算符 ∇（“nabla”）时。其他人（从牛顿到麦克斯韦）一直在为力或速度等矢量的每个分量写出单独的方程。哈密顿本人专注于数学而不是物理，因此他很高兴看到泰特为他的四元数找到了新用途——泰特从贝尔法斯特来到都柏林，以便与哈密顿亲自交谈。他们还交换了数十封信，包括哈密顿的欣喜信，希望有一天他和泰特会因他们的努力而得到感谢。汉密尔顿确实得到了认可，但没有真正感谢他在这个故事中所扮演的角色——但当他于 1865 年去世时，就在麦克斯韦发表开创性的电磁场理论几个月后，34 岁的泰特准备接替他继续工作。

Tait was fascinated by the way quaternions—or rather, the vector part of a quaternion—enabled him to write physics equations so much more compactly, especially when he applied Hamilton’s operator ∇ (“nabla”). Everyone else—from Newton to Maxwell—had been writing out separate equations for each component of a vectorial quantity such as force or velocity. Hamilton himself had focussed on maths rather than physics, so he was delighted by the new uses Tait was finding for his quaternions—and Tait had traveled from Belfast down to Dublin so that he and Hamilton could talk in person. They also exchanged dozens of letters, including Hamilton’s joyful letter hoping that someday he and Tait would be thanked for their efforts. Hamilton did receive recognition but no real thanks for his part in the story—but when he died in 1865, just a few months after Maxwell published his trailblazing electromagnetic field theory, thirtyfour-year-old Tait was ready to carry on in his stead.

首先，他为他的朋友和导师写了一份长达 37 页的讣告——几个月后，也就是 1866 年，它出现在《北不列颠评论》上。他开敦促读者在学术生活中主宰的“争论的喧嚣”和“知识巨人的争斗”中停下来，“在沉思某种更高尚、更微妙的东西中寻求安宁：一个天才的性格和作品。 ”是的，他继续说，“天才”这个词已经被滥用了，因为它指的是比聪明才智更有创造力和独创性的东西——但汉密尔顿是一位最高级别的天才，值得与“所有时代和国家的最伟大天才，如拉格朗日和牛顿”相提并论。因为汉密尔顿的贡献远不止四元数分析，尽管它很了不起。泰特说，他早期关于光学和动力学的论文显示出“对符号的掌握和数学语言的流畅（如果可以使用这种表达方式的话）几乎无人能及。”

First up, he wrote a 37-page obituary for his friend and mentor—it appeared in the North British Review a few months later, in 1866. He opened by urging his readers to pause amid the “din of controversy” and “the battle of intellectual giants” that dominated academic life, and “seek repose in the contemplation of something far more elevated and much more subtle: the character and works of a man of genius.” Yes, he continued, the word “genius” has become overused, because it refers to something far more creative and original than mere cleverness—but Hamilton was a genius of the highest order, fit to rank alongside “those of the grandest of all ages and countries, such as Lagrange and Newton.” For Hamilton contributed far more than quaternion analysis, marvelous though it was. His early papers on optics and dynamics, Tait said, showed “a mastery over symbols, and a flow of mathematical language (if the expression can be used) almost unequalled.”

1867 年，泰特出版了自己的《四元数初等论》，这是第一本（相对）易懂的矢量方法教科书。事实上，除了符号（他效仿汉密尔顿使用希腊字母表示矢量，用S和V代替现代的点和叉表示标量和矢量积）之外，这本书读起来与现代矢量分析教材没什么不同。然而，随着汉密尔顿的离去和新合作者的出现，他应用四元数的热情……并未减弱，因为他在接下来的五年里发表了更多关于这个主题的研究论文，只是有所减弱。因为他的新合作者讨厌四元数！

In 1867, Tait published his own Elementary Treatise on Quaternions, the first (relatively) accessible textbook on vectorial methods. In fact, aside from the notation—he follows Hamilton in using Greek letters for vectors, and S and V instead of the modern dot and cross for scalar and vector products—it reads not unlike a modern vector analysis text. Yet with Hamilton gone and a new collaborator on the scene, his passion for applying quaternions was … not dampened, because he published several more research papers on the subject over the next five years, but somewhat curtailed. For his new collaborator hated quaternions!

我指的是威廉·汤姆森——麦克斯韦的早期导师，未来的开尔文勋爵。他和泰特在 1861 年相识，一年后泰特从贝尔法斯特搬回家乡爱丁堡大学。汤姆森出生于贝尔法斯特，但在格拉斯哥长大，年仅 22 岁时就成为格拉斯哥大学的教授。然而，尽管他发表了一些杰出的数学论文——包括 17 岁时写的关于法拉第场的论文——但汤姆森很快就凭借技术成就而非数学成就而变得富有和出名。例如，在 19 世纪 50 年代末，他曾担任第一条大西洋海底电报电缆铺设工程的首席技术顾问——第一条电缆断裂时他在船上，另一次是一场可怕的风暴，这场风暴威胁着整个项目。最终，电缆铺设完毕，身为北大西洋电报公司董事的汤姆森从欧洲向美国发送了第一批电报信号。他还发明了许多有利可图的电工仪器，并帮助开创了热力学科学——开尔文温标就是以他的名字命名的。在如此多的事情同时进行的情况下，他竟然还能抽出时间与泰特合作，这真是令人惊讶：他们正在编写一本新的物理教科书《自然哲学论》。“自然哲学”仍是理论物理学的名称，他们两人都是这门学科的教授。

I’m speaking of William Thomson—Maxwell’s early mentor, and the future Lord Kelvin. He and Tait had met in 1861, a year after Tait moved from Belfast back home to the University of Edinburgh. Thomson was Belfast-born but Glasgow-raised, and he’d become a professor at the University of Glasgow when he was only twenty-two. Yet despite some remarkable mathematical papers—including the one on Faraday’s field idea, written when he was just seventeen—Thomson would soon become rich and famous for his technical rather than his mathematical achievements. In the late 1850s, for example, he’d been the chief technical advisor during the dramatic laying of the very first telegraph cable under the Atlantic— he’d been on board ship when the first cable broke, and again during a terrible storm that threatened to derail the whole project. Eventually the cable was laid, and Thomson, who was a director of the North Atlantic Telegraph Company, sent the first telegraphic signals from Europe to America. He also invented many profitable electrical instruments and helped pioneer the science of thermodynamics—the Kelvin temperature scale is named after him. With so many irons in the fire, it’s amazing he still managed to find time to collaborate with Tait: they were writing a new physics textbook, A Treatise on Natural Philosophy. “Natural philosophy” was the name still being used for theoretical physics, and it was the subject in which they were both professors.

然而，从 Tait 的角度来看，Thomson 实际上投入到他们的项目上的时间相当少。“我对这本伟大的书感到厌倦，”Tait 在 1864 年写信给他。“如果你只是偶尔给我寄一些零碎的东西……我该怎么办？你甚至没有给我一点提示，说明你想在我们目前关于液体和气体静力学的章节中做什么！”Thomson 当时在德国，Tait 接着说，虽然他几乎完成了关于粒子动力学的一章，但他当然不想支付 45 倍的邮费将其寄到海外，因为 Thomson 很可能要等到回到苏格兰才能读到它^。1

From Tait’s point of view, however, the amount of time Thomson actually devoted to their project was rather small. “I am getting sick of the great Book,” Tait had written to him in 1864. “If you send me only scraps … at rare intervals, what can I do? You have not given me even a hint as to what you want done in our present chapter about statics of liquids and gases!” Thomson was then away in Germany, and Tait went on to say that although he’d almost finished a chapter on the kinetics of particles, he certainly didn’t feel like paying forty-five times more postage to send it overseas when in all probability Thomson wouldn’t get around to reading it till he was back in Scotland.¹

泰特当时还在为他同时撰写的《四元数初等论》写一章关于运动学的内容。他告诉读者，虽然四元数微积分可能很难学，但它值得掌握，因为它“为高级学生提供了最特别的优势，不仅可以帮助他们解决复杂的问题，而且还能提供宝贵的智力训练”。但汤姆森对此不以为然。当然，他使用了矢量分量方程，但他认为根本不需要汉密尔顿的纳布拉算子允许的全矢量微积分。他指出，无论如何你都会用分量进行计算——例如，在标量积中，当你计算$\mathrm{一个}\cdot b={\mathrm{一个}}_1{b}_1+{\mathrm{一个}}_2{b}_2+{\mathrm{一个}}_3{b}_3$或者当你通过计算证明 ∇ ² V = 0$\frac{{\partial }^2五}{\partial {十}^2}+\frac{{\partial }^2五}{\partial {是}^2}+\frac{{\partial }^2五}{\partial {是}^2}$或者在物理和工程问题中，当你将一个向量“分解”为其分量时（如图 8.1所示）——仅举两个例子。因此，对于他们的合作项目，Tait 不情愿地推迟了：四元数被淘汰了。

Tait had also been working on a chapter on kinematics for his Elementary Treatise on Quaternions, which he’d been writing at the same time. He told his readers that although quaternion calculus may be difficult to learn, it is worth mastering because it offers “the most extraordinary advantage to the advanced student, not alone as aiding him in the solution of complex questions, but as affording an invaluable mental discipline.” But Thomson was having none of it. He used vectorial component equations, of course, but he saw no need whatsoever for the whole-vector calculus that Hamilton’s nabla operator allowed. He made the good point that you carried out computations with components anyway—in the scalar product, for example, when you calculate $a\cdot b=a_1{b}_1+a_2{b}_2+a_3{b}_3$. Or when you show that ∇²V = 0 by calculating $\frac{{\partial }^{2}V}{\partial x^2}+\frac{{\partial }^{2}V}{\partial y^2}+\frac{{\partial }^{2}V}{\partial z^2}$, or in physics and engineering problems when you “resolve” a vector into its components (as in fig. 8.1)—to take just two more examples. So, for their joint project, Tait reluctantly deferred: quaternions were banished.

尽管两人意见不一，但汤姆森注意到，泰特总是愿意用莎士比亚、狄更斯和萨克雷的“精彩语录”来“点亮”他们紧张的数学合作。有时，两人会在泰特的小书房里一起工作，煤气灯的灯光在已经被烟雾熏黑的墙上投下怪异的阴影。那些昏暗的墙上还贴着泰特用木炭写下的当时最伟大的科学家名单：汉密尔顿排在第一位，法拉第紧随其后（法拉第于 1867 年去世）。泰特和汤姆森曾讨论过他们的朋友麦克斯韦——用汤姆森的话说，他是“一颗冉冉升起的巨星”——但他们认为他还太年轻，无法进入这份神圣的名单^。2

Despite their disagreements, Thomson noted that Tait was always ready to “brighten” their intense collaborative mathematical labour with “delightful quotations” from Shakespeare, Dickens, and Thackeray. Sometimes the two men worked together in Tait’s small home study, by the light of a gas lamp that cast eerie shadows on a wall already dimmed by tobacco smoke. Those dingy walls also contained a list Tait had written in charcoal of the greatest scientists then living: Hamilton was first, followed by Faraday (who died in 1867). Tait and Thomson had discussed their friend Maxwell—“a rising star of the first magnitude,” as Thomson put it—but thought him still too young to make the hallowed list.²

最后，汤姆森和泰特的《自然哲学论文》终于可以出版了。这本书于 1867 年问世，由于其前沿内容（包括新兴的热力学）而具有巨大影响力。它甚至还讨论了用于复杂数学计算的计算机器。查尔斯·巴贝奇和艾达·洛夫莱斯在 19 世纪 40 年代就提出了这种机器的真正可能性，但巴贝奇的“机器”从未真正投入使用，因此汤姆森描述了他自己的理论设计。这本书被俗称为T和T′（发音为“ T和T -dash”），汤姆森（T）和泰特（T′）的昵称也沿用至今。

At last, Thomson and Tait’s Treatise on Natural Philosophy was ready for the publisher. It saw the light of day in 1867 and would prove very influential because of its cutting-edge content, including the emerging science of thermodynamics. It even included a discussion on calculating machines for sophisticated mathematical computations. Charles Babbage and Ada Lovelace had brought the real possibility of such a machine to attention in the 1840s, but Babbage’s “engine” never quite got off the ground, so Thomson described his own theoretical designs. The book was colloquially known as T and T′—pronounced “T and T-dash”—and the nicknames stuck for Thomson (T) and Tait (T′) themselves, too.

麦克斯韦很快就找到了自己的代号：dp / dt 。这源自泰特 1868 年出版的《热力学概论》一书中热力学第二定律的一种形式：$\frac{d页}{d吨}=J碳米$，其中 JCM 是詹姆斯·克拉克·麦克斯韦的姓名首字母缩写。（我们不需要这个公式，但符号p代表“压力”，J代表“焦耳当量”（热的机械当量），C代表“卡诺函数”，而M与热量有关，今天通常写为dQ / dV。）这很合适，因为麦克斯韦还帮助开创了气体动力学理论——泰特很可能设计了这个方程的形式以表示敬意。麦克斯韦的妻子凯瑟琳曾帮助他进行一些关于热量和颜色的实验，为他点火并调节温度，并进行和记录实验观察。她让我们想起了许多默默无闻的妻子，她们帮助了她们著名的丈夫^。3

Maxwell soon picked up a handle of his own: dp/dt. This came courtesy of a form of the second law of thermodynamics in Tait’s 1868 book, Sketch of Thermodynamics: $\frac{dp}{dt}=JCM$, where JCM are James Clerk Maxwell’s initials. (We don’t need this formula, but the symbol p stands for “pressure,” J for “Joule’s equivalent” (the mechanical equivalent of heat), and C for “Carnot’s function,” while M is related to the amount of heat, usually written today as dQ/dV.) It’s fitting, because Maxwell also helped pioneer the kinetic theory of gases—Tait likely crafted this form of the equation as a tribute. Maxwell’s wife, Katherine, had helped with some of his experiments on heat and colour, keeping the fire stoked and regulating the temperature, and making and recording experimental observations. She reminds us of the many unsung wives who have helped their famous husbands.³

虽然T和T′中没有四元数和全向量微积分，但有大量分量形式方程。 Tait 一定非常乐意在他的《四元数初等论述》中用很小的篇幅重写了其中一些方程。

While quaternions and whole-vector calculus are missing from T and T′, there are plenty of component-form equations. Tait must have had great delight in rewriting some of them, in a fraction of the space, in his Elementary Treatise on Quaternions.

由于其中一些应用涉及电磁学，他立即建议麦克斯韦研究该书的最后二三十页，了解纳布拉的力量。他的信表明麦克斯韦已经对泰特所说的“∇业务”有了一些了解——事实上，麦克斯韦在 1865 年的论文中用 ∇ ²作为拉普拉斯算子的简写。但现在他正计划写一篇关于电磁学的巨作，概述他建立最终理论的所有已知实验和数学结果，然后打算将其与仍然主导主流物理学的超距作用电磁理论进行比较。当他阅读泰特的书时，他对四元数微积分发展中固有的可能性感到兴奋。

Since some of these applications were to electricity and magnetism, he immediately recommended that Maxwell study the last twenty or thirty pages of the book, on the power of nabla. His letter suggests that Maxwell already had some knowledge of this “∇ business,” as Tait put it—and in fact Maxwell had used ∇² as shorthand for the Laplacian operator in his 1865 paper. But now he was planning a monumental treatise on electricity and magnetism, surveying all the known experimental and mathematical results on which he’d built his final theory, which he then intended to compare with the action-at-a-distance electromagnetic theories that still dominated mainstream physics. And as he read Tait’s book, he was excited by the possibilities inherent in its development of quaternion calculus.

事实上，当他在《自然》杂志上评论汤姆森和泰特的“巨作”时，他激动不已，不禁感叹书中甚至没有讨论最简单的矢量规则——尤其是因为其中一位作者是“汉密尔顿的忠实信徒”。当然，汤姆森并不是汉密尔顿的忠实信徒！他认为四元数和矢量微积分不过是已经使用分量设计出来的方程的简写——就像麦克斯韦在 1865 年的论文中建立电磁场方程时所做的那样。但现在，在 19 世纪 70 年代初，麦克斯韦发现了矢量微积分，他告诉泰特，汤姆森似乎没有意识到它是“一把燃烧的剑”，贯穿整个空间，而分量形式则是一把强力的笛卡尔“撞锤，只向西、向北和（向下？）推” ^。4

In fact, he was so excited that when he reviewed Thomson and Tait’s “great Book” for the journal Nature, he couldn’t help but lament the lack of discussion on even the simplest rules of vectors—especially since one of the authors was “an ardent disciple of Hamilton.” Thomson, of course, was not the ardent disciple! He thought quaternion and vector calculus was nothing but a shorthand for equations that had already been devised using components—as Maxwell had done when he built up his electromagnetic field equations in his 1865 paper. But now, in the early 1870s, Maxwell had discovered vector calculus, and he told Tait that Thomson didn’t seem to realise it was “a flaming sword that burns every way”—all through space— whereas the component form was a brute-force Cartesian “ram, pushing [only] westward and northward and (downward?).”⁴

麦克斯韦还向他和泰特的儿时好友刘易斯·坎贝尔通报了最新情况，告诉他“我正在转向四元数”，并提到他在撰写《电磁论》时使用了四元数。他解释说，由于运算符 ∇ 被称为“nabla”，因此它的应用是“Nablody”——麦克斯韦喜欢双关语和胡言乱语！泰特同样爱开玩笑，但他也能更专业，他告诉泰特，他“涉猎汉密尔顿”是因为“我想用汉密尔顿的思想充实我的书，但又不想把这些操作转化为汉密尔顿的形式，因为无论是我还是[,]我认为[,]公众都还没有成熟。现在，汉密尔顿的向量思想的价值是无法形容的”——也就是说，好得无法形容！ ⁵他在一篇新论文《物理量分类评论》中解释了他的意思，这篇论文于 1871 年 3 月 9 日提交给伦敦数学学会。（顺便说一句，在 4 月 13 日举行的下一次学会会议上，会长以一篇简短但热情洋溢、当之无愧的悼词拉开了会议的序幕，悼念德摩根——汉密尔顿的老朋友，现代符号代数的先驱——他于 3 月 18 日去世，享年 64 岁。）

Maxwell also kept his and Tait’s childhood friend Lewis Campbell up to date, telling him, “I am getting converted to Quaternions,” and mentioning that he was using them in the Treatise on Electricity and Magnetism he was writing. He explained that since the operator ∇ was called “nabla,” its application was a “Nablody”—Maxwell loved puns and nonsense! With Tait he was equally playful, but he was also able to be much more technical, and he’d told him he was “dabbling in Hamilton” because “I want to leaven my book with Hamiltonian ideas without casting the operations into Hamiltonian form for which neither I nor[,] I think[,] the public are ripe. Now the value of Hamilton’s idea of a vector is unspeakable”—that is, unspeakably good!⁵ He explained what he meant by this in a new paper, “Remarks on the Classification of Physical Quantities,” which he presented to the London Mathematical Society on March 9, 1871. (Incidentally, at the very next meeting of the society, on April 13, the president opened the proceedings with a brief but glowing and well-deserved elegy for De Morgan—Hamilton’s old friend, and a pioneer of modern symbolic algebra—who had died on March 18, aged sixty-four.)

首先，麦克斯韦谈到了对随着科学进步而产生的大量物理量进行分类的必要性，并指出如果通过数学类比进行分类，那么对于解决方程形式与已知问题相同的新问题将大有裨益。我们之前已经看到，振动弦与麦克斯韦的电磁波之间存在波动方程类比，汤姆逊利用傅立叶热流分析来类比描述法拉第的场线或“力线”，以及麦克斯韦类似的流线类比。麦克斯韦提到了汤姆逊，但他接着说，汉密尔顿通过将物理量分为标量和矢量，提供了一种比类比更基本的分类方法。麦克斯韦说，这一创新成果卓著，四元数的重要性只有笛卡尔发明的坐标系才能与之媲美。他的观点非常正确，因为四元数和向量（及其张量扩展）是现代物理学家、工程师和数据科学家描述和分析物体占据物理和数字空间的方式的基础。回到过去并“见证”一个全新想法（如汉密尔顿的想法）的诞生，并事后观察谁意识到了它的重要性，谁没有意识到，这是一件非常有趣的事情。

First Maxwell spoke of the need to classify the burgeoning number of physical quantities that had arisen because of the progress of science— pointing out that if this classification is done through mathematical analogy, it can be of great help in solving new problems whose equations have the same form as that of a known problem. We saw this earlier with the wave equation analogy between vibrating strings and Maxwell’s electromagnetic waves, and with Thomson’s use of Fourier’s analysis of heat flow as an analogy for describing Faraday’s field lines, or “lines of force,” and with Maxwell’s similar streamline analogy. Maxwell mentioned Thomson, but he went on to say that Hamilton had provided an even more fundamental type of classification than analogy, by dividing physical quantities into scalars and vectors. This innovation was so fruitful, Maxwell said, that the importance of quaternions could only be compared with Descartes’s invention of coordinates. He was spot on, given that quaternions and vectors (and their tensor extensions) are fundamental to the way modern physicists, engineers, and data scientists describe and analyse the way objects occupy physical and digital spaces. It’s fascinating to go back in time and “witness” a brand-new idea, such as Hamilton’s, as it comes into being—and to observe, with hindsight, just who did and who didn’t recognise its significance.

麦克斯韦接着定义了两个新的矢量子类别：力和通量。他说，力是与长度或距离相关的矢量（如线积分），而通量（如磁感应）是与长度或距离相关的矢量到面积或曲面积分。正如我在第 6 章中概述的那样，在他 1865 年的论文中，他使用了散度定理和斯托克斯定理的分量形式将这些积分与导数联系起来。然而，到 1871 年，他发现了哈密顿 nabla 算子 ∇ 的强大功能，它可以用整个矢量和矢量场来定义这些物理量。我们很快就会看到这些 nabla 运算，它们非常重要，麦克斯韦说，他给它们起了特殊的名字。你可能已经知道它们：“散度”、“旋度”和“梯度”——但麦克斯韦把它们给了我们，所以值得花点时间看看他为什么选择它们。

Maxwell went on to define two new subcategories of vectors: forces and fluxes. Forces, he said, are vectors related to length or distance—as in line integrals—and fluxes (such as magnetic induction) are vectors related to areas, or surface integrals. For his 1865 paper he’d used the component form of the divergence theorem and Stokes’s theorem to relate these integrals to derivatives, as I outlined in chapter 6. By 1871, however, he’d discovered the power of Hamilton’s nabla operator, ∇, for defining such physical quantities in terms of whole vectors and vector fields. We’ll see these nabla operations shortly, and they are so important, Maxwell said, that he gave them special names. You might already know them: “divergence,” “curl,” and “grad”—but it was Maxwell who gave them to us, so it’s worth spending a little time seeing why he chose them.

首先，他建议，“将 ∇ 应用于标量函数的结果可以称为斜率。”这与斜率类似，$米=\frac{d是}{d十}$，直线y = mx + c的函数，因为当你将 ∇ 应用于标量函数（例如电势（或电压）或力的势）时，同样需要求导。我们通常认为标量就是数字，但如果函数的数值不依赖于你使用的坐标，那么它们就可以是标量。再举一个例子，压力和温度是标量，因此压力和/或温度的函数是标量函数。无论如何，正如我在第 6 章中展示的那样，F = ∇ V（及其现代矢量等价物F = ∇ V）意味着力具有分量$\frac{\partial 五}{\partial 十}，\frac{\partial 五}{\partial 是}，\frac{\partial 五}{\partial 是}，$类似于直线斜率$\frac{d是}{d十}$今天我们大声朗读F = ∇ V为“ F等于 grad V ”，其中“grad”是“gradient”的缩写，是“slope”的另一个名称。所以，麦克斯韦的术语确实有点道理！它显示了标量函数在空间中的变化方式。

First up, he suggested, “the result of ∇ applied to a scalar function might be called the slope.” That’s by analogy with the slope, $m=\frac{dy}{dx}$, of a straight line y = mx + c, because similarly you’re taking derivatives when you apply ∇ to a scalar function, such as the electric potential (or voltage), or the potential of a force. We often think of scalars as just numbers, but functions can be scalars if they take numerical values that don’t depend on the coordinates you’re using. To take another example, pressure and temperature are scalars, so a function of pressure and/or temperature is a scalar function. Anyway, as I showed in chapter 6, F = ∇V (and its modern vector equivalent F = ∇V ) means that the force has components $\frac{\partial V}{\partial x},\frac{\partial V}{\partial y},\frac{\partial V}{\partial z},$ analogous to the straight-line slope $\frac{dy}{dx}$. Today we read F = ∇V aloud as “F equals grad V,” where “grad” is short for “gradient,” which is another name for “slope.” So, Maxwell’s term did sort of stick! It shows how a scalar function changes through space.